1.
Modular arithmetic
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In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, a familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7,00 now, then 8 hours later it will be 3,00. Usual addition would suggest that the time should be 7 +8 =15. Likewise, if the clock starts at 12,00 and 21 hours elapse, then the time will be 9,00 the next day, because the hour number starts over after it reaches 12, this is arithmetic modulo 12. According to the definition below,12 is congruent not only to 12 itself, Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations on integers, addition, subtraction, and multiplication. For a positive n, two integers a and b are said to be congruent modulo n, written, a ≡ b. The number n is called the modulus of the congruence, for example,38 ≡14 because 38 −14 =24, which is a multiple of 12. The same rule holds for negative values, −8 ≡72 ≡ −3 −3 ≡ −8. Equivalently, a ≡ b mod n can also be thought of as asserting that the remainders of the division of both a and b by n are the same, for instance,38 ≡14 because both 38 and 14 have the same remainder 2 when divided by 12. It is also the case that 38 −14 =24 is a multiple of 12. A remark on the notation, Because it is common to consider several congruence relations for different moduli at the same time, in spite of the ternary notation, the congruence relation for a given modulus is binary. This would have been if the notation a ≡n b had been used. The properties that make this relation a congruence relation are the following, if a 1 ≡ b 1 and a 2 ≡ b 2, then, a 1 + a 2 ≡ b 1 + b 2 a 1 − a 2 ≡ b 1 − b 2. The above two properties would still hold if the theory were expanded to all real numbers, that is if a1, a2, b1, b2. The next property, however, would fail if these variables were not all integers, the notion of modular arithmetic is related to that of the remainder in Euclidean division. The operation of finding the remainder is referred to as the modulo operation. For example, the remainder of the division of 14 by 12 is denoted by 14 mod 12, as this remainder is 2, we have 14 mod 12 =2
2.
Coprime integers
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In number theory, two integers a and b are said to be relatively prime, mutually prime, or coprime if the only positive integer that divides both of them is 1. That is, the common positive factor of the two numbers is 1. This is equivalent to their greatest common divisor being 1, the numerator and denominator of a reduced fraction are coprime. In addition to gcd =1 and =1, the notation a ⊥ b is used to indicate that a and b are relatively prime. For example,14 and 15 are coprime, being divisible by only 1. The numbers 1 and −1 are the only integers coprime to every integer, a fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. The number of integers coprime to an integer n, between 1 and n, is given by Eulers totient function φ. A set of integers can also be called if its elements share no common positive factor except 1. A set of integers is said to be pairwise coprime if a and b are coprime for every pair of different integers in it, a number of conditions are individually equivalent to a and b being coprime, No prime number divides both a and b. There exist integers x and y such that ax + by =1, the integer b has a multiplicative inverse modulo a, there exists an integer y such that by ≡1. In other words, b is a unit in the ring Z/aZ of integers modulo a, the least common multiple of a and b is equal to their product ab, i. e. LCM = ab. As a consequence of the point, if a and b are coprime and br ≡ bs. That is, we may divide by b when working modulo a, as a consequence of the first point, if a and b are coprime, then so are any powers ak and bl. If a and b are coprime and a divides the product bc and this can be viewed as a generalization of Euclids lemma. In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6/π2, which is about 61%, two natural numbers a and b are coprime if and only if the numbers 2a −1 and 2b −1 are coprime. As a generalization of this, following easily from the Euclidean algorithm in base n >1, a set of integers S = can also be called coprime or setwise coprime if the greatest common divisor of all the elements of the set is 1. For example, the integers 6,10,15 are coprime because 1 is the positive integer that divides all of them. If every pair in a set of integers is coprime, then the set is said to be pairwise coprime, pairwise coprimality is a stronger condition than setwise coprimality, every pairwise coprime finite set is also setwise coprime, but the reverse is not true
3.
Group (mathematics)
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In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
4.
Ring (mathematics)
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In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices, the conceptualization of rings started in the 1870s and completed in the 1920s. Key contributors include Dedekind, Hilbert, Fraenkel, and Noether, rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry. A ring is a group with a second binary operation that is associative, is distributive over the abelian group operation. By extension from the integers, the group operation is called addition. Whether a ring is commutative or not has profound implications on its behavior as an abstract object, as a result, commutative ring theory, commonly known as commutative algebra, is a key topic in ring theory. Its development has greatly influenced by problems and ideas occurring naturally in algebraic number theory. The most familiar example of a ring is the set of all integers, Z, −5, −4, −3, −2, −1,0,1,2,3,4,5. The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings, a ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms 1. R is a group under addition, meaning that, + c = a + for all a, b, c in R. a + b = b + a for all a, b in R. There is an element 0 in R such that a +0 = a for all a in R, for each a in R there exists −a in R such that a + =0. R is a monoid under multiplication, meaning that, · c = a · for all a, b, c in R. There is an element 1 in R such that a ·1 = a and 1 · a = a for all a in R.3. Multiplication is distributive with respect to addition, a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many follow a alternative convention in which a ring is not defined to have a multiplicative identity. This article adopts the convention that, unless stated, a ring is assumed to have such an identity
5.
Abstract algebra
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In algebra, which is a broad division of mathematics, abstract algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, the term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra. Algebraic structures, with their homomorphisms, form mathematical categories. Category theory is a formalism that allows a way for expressing properties. Universal algebra is a subject that studies types of algebraic structures as single objects. For example, the structure of groups is an object in universal algebra. As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra, through the end of the nineteenth century, many – perhaps most – of these problems were in some way related to the theory of algebraic equations. Numerous textbooks in abstract algebra start with definitions of various algebraic structures. This creates an impression that in algebra axioms had come first and then served as a motivation. The true order of development was almost exactly the opposite. For example, the numbers of the nineteenth century had kinematic and physical motivations. An archetypical example of this progressive synthesis can be seen in the history of group theory, there were several threads in the early development of group theory, in modern language loosely corresponding to number theory, theory of equations, and geometry. Leonhard Euler considered algebraic operations on numbers modulo an integer, modular arithmetic, lagranges goal was to understand why equations of third and fourth degree admit formulae for solutions, and he identified as key objects permutations of the roots. An important novel step taken by Lagrange in this paper was the view of the roots, i. e. as symbols. However, he did not consider composition of permutations, serendipitously, the first edition of Edward Warings Meditationes Algebraicae appeared in the same year, with an expanded version published in 1782. Waring proved the theorem on symmetric functions, and specially considered the relation between the roots of a quartic equation and its resolvent cubic. Kronecker claimed in 1888 that the study of modern algebra began with this first paper of Vandermonde, cauchy states quite clearly that Vandermonde had priority over Lagrange for this remarkable idea, which eventually led to the study of group theory. Paolo Ruffini was the first person to develop the theory of permutation groups and his goal was to establish the impossibility of an algebraic solution to a general algebraic equation of degree greater than four
6.
Group theory
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra, linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is central to public key cryptography. The first class of groups to undergo a systematic study was permutation groups, given any set X and a collection G of bijections of X into itself that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn, in general, an early construction due to Cayley exhibited any group as a permutation group, acting on itself by means of the left regular representation. In many cases, the structure of a group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥5 and this fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥5 in radicals. The next important class of groups is given by matrix groups, here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the vector space Kn by linear transformations. In the case of groups, X is a set, for matrix groups. The concept of a group is closely related with the concept of a symmetry group. The theory of groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, the groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were concrete, having been realized through numbers, permutations, or matrices. It was not until the nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, a significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory
7.
Subgroup
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In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is an operation on H. This is usually denoted H ≤ G, read as H is a subgroup of G, the trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group G is a subgroup H which is a subset of G. This is usually represented notationally by H < G, read as H is a subgroup of G. Some authors also exclude the group from being proper. If H is a subgroup of G, then G is sometimes called an overgroup of H, the same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G is sometimes denoted by the pair, usually to emphasize the operation ∗ when G carries multiple algebraic or other structures. This article will write ab for a ∗ b, as is usual, a subset H of the group G is a subgroup of G if and only if it is nonempty and closed under products and inverses. In the case that H is finite, then H is a subgroup if and only if H is closed under products. The above condition can be stated in terms of a homomorphism, the identity of a subgroup is the identity of the group, if G is a group with identity eG, and H is a subgroup of G with identity eH, then eH = eG. The intersection of subgroups A and B is again a subgroup. The union of subgroups A and B is a if and only if either A or B contains the other, since for example 2 and 3 are in the union of 2Z and 3Z. Another example is the union of the x-axis and the y-axis in the plane, each of these objects is a subgroup and this also serves as an example of two subgroups, whose intersection is precisely the identity. An element of G is in <S> if and only if it is a product of elements of S. Every element a of a group G generates the cyclic subgroup <a>, if <a> is isomorphic to Z/nZ for some positive integer n, then n is the smallest positive integer for which an = e, and n is called the order of a. If <a> is isomorphic to Z, then a is said to have infinite order, the subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. If e is the identity of G, then the group is the minimum subgroup of G
8.
Normal subgroup
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In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup H of a group G is normal in G if and only if gH = Hg for all g in G, i. e. the sets of left, Normal subgroups can be used to construct quotient groups from a given group. Évariste Galois was the first to realize the importance of the existence of normal subgroups, for any subgroup, the following conditions are equivalent to normality. Therefore, any one of them may be taken as the definition, the image of conjugation of N by any element of G is a subset of N, ∀g ∈ G, gNg−1 ⊆ N. The image of conjugation of N by any element of G is N, ∀g ∈ G, the sets of left and right cosets of N in G coincide, ∀g ∈ G, gN = Ng. N is a union of conjugacy classes of G, N = ⋃g∈N Cl, there is some homomorphism on G for which N is the kernel, ∃φ ∈ Hom ∣ ker φ = N. The last condition accounts for some of the importance of normal subgroups, the subgroup consisting of just the identity element of G and G itself are always normal subgroups of G. The former is called the trivial subgroup, and if these are the normal subgroups. The center of a group is a normal subgroup, the commutator subgroup is a normal subgroup. More generally, any characteristic subgroup is normal, since conjugation is always an automorphism, all subgroups, N, of an abelian group, G, are normal, because gN = Ng. A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group, the translations by a given distance in any direction form a conjugacy class, the translation group is the union of those for all distances. Normality is preserved upon surjective homomorphisms, and is preserved upon taking inverse images. Normality is preserved on taking direct products, if H is a normal subgroup of G, and K is a subgroup of G containing H, then H is a normal subgroup of K. A normal subgroup of a subgroup of a group need not be normal in the group. That is, normality is not a transitive relation, the smallest group exhibiting this phenomenon is the dihedral group of order 8. However, a subgroup of a normal subgroup is normal. Also, a subgroup of a central factor is normal. In particular, a subgroup of a direct factor is normal
9.
Quotient group
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A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure. It is part of the field known as group theory. The resulting quotient is written G / N, where G is the original group, much of the importance of quotient groups is derived from their relation to homomorphisms. The dual notion of a quotient group is a subgroup, these being the two ways of forming a smaller group from a larger one. Any normal subgroup has a quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In category theory, quotient groups are examples of quotient objects, for other examples of quotient objects, see quotient ring, quotient space, quotient space, and quotient set. Given a group G and a subgroup H, and an element a in G, then one can consider the left coset, aH. Cosets are a class of subsets of a group, for example consider the abelian group G of integers, with operation defined by the usual addition. Then there are exactly two cosets,0 + H, which are the integers, and 1 + H. For a general subgroup H, it is desirable to define a group operation on the set of all possible cosets. This is possible exactly when H is a subgroup, as we will see below. A subgroup N of a group G is normal if and only if the coset equality aN = Na holds for all a in G, a normal subgroup of G is denoted N ◁ G. Let N be a subgroup of a group G. We define the set G/N to be the set of all cosets of N in G, i. e. G/N =. Define an operation on G/N as follows, for each aN and bN in G/N, the product of aN and bN is. This defines an operation on G/N if we impose = aN = aN = NN = N, here we have used in an important way that N is a normal subgroup. One checks that this operation on G/N is associative, has identity element N, therefore, the set G/N together with the operation defined above forms a group, this is known as the quotient group of G by N. Because of the normality of N, the left cosets and right cosets of N in G are equal, for example, consider the group with addition modulo 6, G =
10.
Semidirect product
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In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. As with direct products, there is an equivalence between inner and outer semidirect products, and both are commonly referred to simply as semidirect products. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product, for every g ∈ G, there are unique n ∈ N and h ∈ H, such that g = nh. For every g ∈ G, there are unique h ∈ H and n ∈ N, such that g = hn. The composition π ∘ i of the embedding i, H → G. There exists a homomorphism G → H that is the identity on H, to avoid ambiguity, it is advisable to specify which is the normal subgroup. Let G be a product of the normal subgroup N. Let Aut denote the group of all automorphisms of N, the map φ, H → Aut defined by φ = φh, conjugation by h, where φ = φh = hnh−1 for all h in H and n in N, is a group homomorphism. Together N, H, and φ determine G up to isomorphism, as we show now. Given any two groups N and H and a group homomorphism φ, H → Aut, we can construct a new group N ⋊φ H, called the product of N and H with respect to φ. This defines a group in which the identity element is and the inverse of the element is, pairs form a normal subgroup isomorphic to N, while pairs form a subgroup isomorphic to H. The full group is a product of those two subgroups in the sense given earlier. Let φ, H → Aut be the homomorphism given by φ h = h n h −1 for all n ∈ N, h ∈ H. Then G is isomorphic to the semidirect product N ⋊φ H, and applying the isomorphism to the product, nh, gives the tuple. In G, we have = n 1 h 1 n 2 h 2 = = ∙ which shows that the map is indeed an isomorphism. The direct product is a case of the semidirect product. To see this, let φ be the trivial homomorphism then N ⋊φ H is the direct product N × H, in this case, φ, H → Aut is given by φ = φh, where φ h = β −1. The dihedral group D2n with 2n elements is isomorphic to a product of the cyclic groups Cn
11.
Direct product of groups
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In group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of groups, the direct product is sometimes referred to as the direct sum. Given groups G and H, the direct product G × H is defined as follows, Specifically, Associativity The binary operation on G × H is indeed associative. If we identify these with G and H, respectively, then we can think of the direct product P as containing the original groups G and H as subgroups and these subgroups of P have the following three important properties, The intersection G ∩ H is trivial. Every element of P can be expressed as the product of an element of G, every element of G commutes with every element of H. Together, these three properties completely determine the algebraic structure of the direct product P. That is, if P is any group having subgroups G and H that satisfy the properties above, in this situation, P is sometimes referred to as the internal direct product of its subgroups G and H. In some contexts, the property above is replaced by the following. Both G and H are normal in P, then G × H = 〈 a, b | a3 =1, b5 =1, ab = ba 〉. As mentioned above, the subgroups G and H are normal in G × H. Specifically, define functions πG, G × H → G and πH, G × H → H by πG = g and πH = h. Then πG and πH are homomorphisms, known as projection homomorphisms, whose kernels are H and G and it follows that G × H is an extension of G by H. In the case where G × H is a group, it follows that the composition factors of G × H are precisely the union of the composition factors of G. The direct product G × H can be characterized by the universal property. Let πG, G × H → G and πH, G × H → H be the projection homomorphisms and this is a special case of the universal property for products in category theory. If A is a subgroup of G and B is a subgroup of H, for example, the isomorphic copy of G in G × H is the product G ×, where is the trivial subgroup of H. If A and B are normal, then A × B is a subgroup of G × H. Moreover. Note that it is not true in general that every subgroup of G × H is the product of a subgroup of G with a subgroup of H. For example, if G is any group, then the product G × G has a diagonal subgroup Δ = which is not the product of two subgroups of G
12.
Group homomorphism
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From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that h = h −1. Hence one can say that h is compatible with the group structure, older notations for the homomorphism h may be xh, though this may be confused as an index or a general subscript. A more recent trend is to write group homomorphisms on the right of their arguments, omitting brackets and this approach is especially prevalent in areas of group theory where automata play a role, since it accords better with the convention that automata read words from left to right. In areas of mathematics where one considers groups endowed with additional structure, for example, a homomorphism of topological groups is often required to be continuous. The purpose of defining a group homomorphism is to create functions that preserve the algebraic structure, an equivalent definition of group homomorphism is, The function h, G → H is a group homomorphism if whenever a ∗ b = c we have h ⋅ h = h. In other words, the group H in some sense has an algebraic structure as G. Monomorphism A group homomorphism that is injective, i. e. preserves distinctness, epimorphism A group homomorphism that is surjective, i. e. reaches every point in the codomain. Isomorphism A group homomorphism that is bijective, i. e. injective and surjective and its inverse is also a group homomorphism. In this case, the groups G and H are called isomorphic, endomorphism A homomorphism, h, G → G, the domain and codomain are the same. Also called an endomorphism of G. Automorphism An endomorphism that is bijective, the set of all automorphisms of a group G, with functional composition as operation, forms itself a group, the automorphism group of G. As an example, the group of contains only two elements, the identity transformation and multiplication with −1, it is isomorphic to Z/2Z. We define the kernel of h to be the set of elements in G which are mapped to the identity in H ker ≡. the kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The first isomorphism theorem states that the image of a group homomorphism, if and only if ker =, the homomorphism, h, is a group monomorphism, i. e. h is injective. The map h, Z → Z/3Z with h = u mod 3 is a group homomorphism and it is surjective and its kernel consists of all integers which are divisible by 3. The exponential map yields a homomorphism from the group of real numbers R with addition to the group of non-zero real numbers R* with multiplication. The kernel is and the image consists of the real numbers. The exponential map yields a group homomorphism from the group of complex numbers C with addition to the group of non-zero complex numbers C* with multiplication. This map is surjective and has the kernel, as can be seen from Eulers formula, fields like R and C that have homomorphisms from their additive group to their multiplicative group are thus called exponential fields
13.
Image (mathematics)
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In mathematics, an image is the subset of a functions codomain which is the output of the function from a subset of its domain. Evaluating a function at each element of a subset X of the domain, the inverse image or preimage of a particular subset S of the codomain of a function is the set of all elements of the domain that map to the members of S. Image and inverse image may also be defined for binary relations. The word image is used in three related ways, in these definitions, f, X → Y is a function from the set X to the set Y. If x is a member of X, then f = y is the image of x under f, Y is alternatively known as the output of f for argument x. The image of a subset A ⊆ X under f is the subset f ⊆ Y defined by, f = When there is no risk of confusion and this convention is a common one, the intended meaning must be inferred from the context. This makes the image of f a function whose domain is the set of X. The image f of the entire domain X of f is called simply the image of f, let f be a function from X to Y. The set of all the fibers over the elements of Y is a family of sets indexed by Y, for example, for the function f = x2, the inverse image of would be. Again, if there is no risk of confusion, we may denote f −1 by f −1, the notation f −1 should not be confused with that for inverse function. The notation coincides with the one, though, for bijections. The traditional notations used in the section can be confusing. {\displaystyle f=\left\ The image of the set under f is f =, the image of the function f is. The preimage of a is f −1 =, the preimage of is the empty set. F, R → R defined by f = x2, the image of under f is f =, and the image of f is R+. The preimage of f is f −1 =. The preimage of set N = under f is the empty set, F, R2 → R defined by f = x2 + y2. The fibres f −1 are concentric circles about the origin, the origin itself, and the empty set, depending on whether a >0, a =0, or a <0, respectively
14.
Direct sum of groups
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If G is the direct sum of subgroups H and K, then we write G = H + K, if G is the direct sum of a set of subgroups, we often write G = ∑Hi. Loosely speaking, a sum is isomorphic to a weak direct product of subgroups. In abstract algebra, this method of construction can be generalized to direct sums of vector spaces, modules and this notation is commutative, so that in the case of the direct sum of two subgroups, G = H + K = K + H. It is also associative in the sense that if G = H + K, a group which can be expressed as a direct sum of non-trivial subgroups is called decomposable, otherwise it is called indecomposable. If i ≠ j, then for all hi in Hi, hj in Hj, we have that hi * hj = hj * hi for each g in G, there unique set of such that g = h1*h2*. If we take G = ∏ i ∈ I H i it is clear that G is the product of the subgroups H i 0 × ∏ i ≠ i 0 H i. If H is a subgroup of an abelian group G. To describe the properties in the case where G is the direct sum of an infinite set of subgroups. If g is an element of the cartesian product ∏ of a set of groups, let gi be the ith element of g in the product. The external direct sum of a set of groups is the subset of ∏, the group operation in the external direct sum is pointwise multiplication, as in the usual direct product. This subset does indeed form a group, and for a set of groups Hi. If G = ∑Hi, then G is isomorphic to ∑E. Thus, in a sense, for each element g in G, there is a unique finite set S and unique such that g = ∏. Direct sum coproduct free product Direct sum of topological groups
15.
Wreath product
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In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product. Wreath products are used in the classification of groups and also provide a way of constructing interesting examples of groups. Given two groups A and H, there exist two variations of the product, the unrestricted wreath product A Wr H and the restricted wreath product A wr H. Given a set Ω with an H-action there exists a generalisation of the product which is denoted by A WrΩ H or A wrΩ H respectively. The notion generalizes to semigroups and is a construction in the Krohn-Rhodes structure theory of finite semigroups. Let A and H be groups and Ω a set with H acting on it, let K be the direct product K ≡ ∏ ω ∈ Ω A ω of copies of Aω, = A indexed by the set Ω. The elements of K can be seen as arbitrary sequences of elements of A indexed by Ω with component wise multiplication, then the action of H on Ω extends in a natural way to an action of H on the group K by h ≡. Then the unrestricted wreath product A WrΩ H of A by H is the semidirect product K ⋊ H, the subgroup K of A WrΩ H is called the base of the wreath product. The restricted wreath product A wrΩ H is constructed in the way as the unrestricted wreath product except that one uses the direct sum K ≡ ⨁ ω ∈ Ω A ω as the base of the wreath product. In this case the elements of K are sequences of elements in A indexed by Ω of which all, in the most common case, one takes Ω, = H, where H acts in a natural way on itself by left multiplication. In this case, the unrestricted and restricted wreath product may be denoted by A Wr H and A wr H respectively and this is called the regular wreath product. The structure of the product of A by H depends on the H-set Ω. However, in literature the notation used may be deficient and one needs to pay attention on the circumstances, in literature the H-set Ω may be omitted from the notation even if Ω≠H. In the special case that H = Sn is the group of degree n it is common in the literature to assume that Ω=. That is, A≀Sn commonly denotes A≀Sn instead of the wreath product A≀SnSn. In the first case the group is the product of n copies of A. Since the finite direct product is the same as the direct sum of groups, it follows that the unrestricted A WrΩ H. In particular this is true when Ω = H is finite, a wrΩ H is always a subgroup of A WrΩ H
16.
Simple group
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In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two groups, a normal subgroup and the quotient group, and the process can be repeated. If the group is finite, then one arrives at uniquely determined simple groups by the Jordan–Hölder theorem. The complete classification of simple groups, completed in 2008, is a major milestone in the history of mathematics. The cyclic group G = Z/3Z of congruence classes modulo 3 is simple, If H is a subgroup of this group, its order must be a divisor of the order of G which is 3. Since 3 is prime, its only divisors are 1 and 3, so either H is G, on the other hand, the group G = Z/12Z is not simple. The set H of congruence classes of 0,4, and 8 modulo 12 is a subgroup of order 3, similarly, the additive group Z of integers is not simple, the set of even integers is a non-trivial proper normal subgroup. One may use the kind of reasoning for any abelian group. The classification of simple groups is far less trivial. The smallest nonabelian group is the alternating group A5 of order 60. The second smallest nonabelian group is the projective special linear group PSL of order 168. The infinite alternating group, i. e. the group of permutations of the integers. This group can be defined as the union of the finite simple groups A n with respect to standard embeddings A n → A n +1. Another family of examples of simple groups is given by P S L n. It is much more difficult to construct finitely generated infinite simple groups, the first example is due to Graham Higman and is a quotient of the Higman group. Other examples include the infinite Thompson groups T and V. Finitely presented torsion-free infinite simple groups were constructed by Burger-Mozes, there is as yet no known classification for general simple groups. This is expressed by the Jordan–Hölder theorem which states that any two composition series of a group have the same length and the same factors, up to permutation. In a huge effort, the classification of finite simple groups was declared accomplished in 1983 by Daniel Gorenstein
17.
Finite group
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In abstract algebra, a finite group is a mathematical group with a finite number of elements. A group is a set of elements together with an operation which associates, to each ordered pair of elements, with a finite group, the set is finite. As a consequence, the classification of finite simple groups was achieved. During the second half of the century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups. One such family of groups is the family of linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical or physical objects, the theory of Lie groups, which may be viewed as dealing with continuous symmetry, is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite-dimensional Euclidean space, the properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry. Since there are n. possible permutations of a set of n symbols, a cyclic group Zn is a group all of whose elements are powers of a particular element a where an = a0 = e, the identity. A typical realization of this group is as the nth roots of unity. Sending a to a root of unity gives an isomorphism between the two. This can be done with any finite cyclic group, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order. They are named after Niels Henrik Abel, an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of an abelian group can be described directly in terms of these invariants. A group of Lie type is a closely related to the group G of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type give the bulk of nonabelian simple groups. Special cases include the groups, the Chevalley groups, the Steinberg groups. The systematic exploration of finite groups of Lie type started with Camille Jordans theorem that the special linear group PSL is simple for q ≠2,3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL of finite simple groups, other classical groups were studied by Leonard Dickson in the beginning of 20th century
18.
Topological group
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A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. A topological group, G, is a space which is also a group such that the group operations of product, G × G → G, ↦ x y and taking inverses. Here G × G is viewed as a space with the product topology. Although not part of this definition, many require that the topology on G be Hausdorff. The reasons, and some equivalent conditions, are discussed below, in any case, any topological group can be made Hausdorff by taking an appropriate canonical quotient. Note that the axioms are given in terms of the maps, a homomorphism of topological groups means a continuous group homomorphism G → H. An isomorphism of groups is a group isomorphism which is also a homeomorphism of the underlying topological spaces. This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous, there are examples of topological groups which are isomorphic as ordinary groups but not as topological groups. Indeed, any topological group is also a topological group when considered with the discrete topology. The underlying groups are the same, but as topological groups there is not an isomorphism, Topological groups, together with their homomorphisms, form a category. Every group can be made into a topological group by considering it with the discrete topology. In this sense, the theory of topological groups subsumes that of ordinary groups, the real numbers, R with the usual topology form a topological group under addition. More generally, Euclidean n-space Rn is a group under addition. Some other examples of topological groups are the circle group S1. The classical groups are important examples of topological groups. Another classical group is the orthogonal group O, the group of all maps from Rn to itself that preserve the length of all vectors. The orthogonal group is compact as a topological space, much of Euclidean geometry can be viewed as studying the structure of the orthogonal group, or the closely related group O ⋉ Rn of isometries of Rn
19.
Multiplicative group
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In the case of a field F, the group is, where 0 refers to the zero element of F and the binary operation • is the field multiplication, the algebraic torus GL. The multiplicative group of integers modulo n is the group multiplication of the invertible elements of Z / n Z. When n is not prime, there are other than zero that are not invertible. The multiplicative group of real numbers, R +, is an abelian group with 1 being its neutral element. The logarithm is an isomorphism of this group to the additive group of real numbers. The group scheme of n-th roots of unity is by definition the kernel of the map on the multiplicative group GL. The resulting group scheme is written μn and it gives rise to a reduced scheme, when we take it over a field K, if and only if the characteristic of K does not divide n. This makes it a source of some key examples of non-reduced schemes and this phenomenon is not easily expressed in the classical language of algebraic geometry. It turns out to be of importance, for example. The Galois cohomology of this scheme is a way of expressing Kummer theory. Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko, ISBN 1-4020-2690-0 Multiplicative group of integers modulo n Additive group
20.
Cyclic group
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In algebra, a cyclic group or monogenous group is a group that is generated by a single element. Each element can be written as a power of g in multiplicative notation and this element g is called a generator of the group. Every infinite cyclic group is isomorphic to the group of Z. Every finite cyclic group of n is isomorphic to the additive group of Z/nZ. Every cyclic group is a group, and every finitely generated abelian group is a direct product of cyclic groups. A group G is called if there exists an element g in G such that G = ⟨g⟩ =. Since any group generated by an element in a group is a subgroup of that group, for example, if G = is a group of order 6, then g6 = g0, and G is cyclic. In fact, G is essentially the same as the set with addition modulo 6, for example,1 +2 ≡3 corresponds to g1 · g2 = g3, and 2 +5 ≡1 corresponds to g2 · g5 = g7 = g1, and so on. One can use the isomorphism χ defined by χ = i, the name cyclic may be misleading, it is possible to generate infinitely many elements and not form any literal cycles, that is, every gn is distinct. A group generated in this way is called a cyclic group. The French mathematicians known as Nicolas Bourbaki referred to a group as a monogenous group. The set of integers, with the operation of addition, forms a group and it is an infinite cyclic group, because all integers can be written as a finite sum or difference of copies of the number 1. In this group,1 and −1 are the only generators, every infinite cyclic group is isomorphic to this group. For every positive n, the set of integers modulo n, again with the operation of addition, forms a finite cyclic group. An element g is a generator of this group if g is relatively prime to n, thus, the number of different generators is φ, where φ is the Euler totient function, the function that counts the number of numbers modulo n that are relatively prime to n. Every finite cyclic group is isomorphic to a group Z/n, where n is the order of the group, the integer and modular addition operations, used to define the cyclic groups, are both the addition operations of commutative rings, also denoted Z and Z/n. If p is a prime, then Z/p is a finite field, every field with p elements is isomorphic to this one. For every positive n, the subset of the integers modulo n that are relatively prime to n, with the operation of multiplication
21.
Abelian group
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That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers and they are named after Niels Henrik Abel. The concept of a group is one of the first concepts encountered in undergraduate abstract algebra, from which many other basic concepts, such as modules. The theory of groups is generally simpler than that of their non-abelian counterparts. On the other hand, the theory of abelian groups is an area of current research. An abelian group is a set, A, together with an operation • that combines any two elements a and b to form another element denoted a • b, the symbol • is a general placeholder for a concretely given operation. Identity element There exists an element e in A, such that for all elements a in A, the equation e • a = a • e = a holds. Inverse element For each a in A, there exists an element b in A such that a • b = b • a = e, commutativity For all a, b in A, a • b = b • a. A group in which the operation is not commutative is called a non-abelian group or non-commutative group. There are two main conventions for abelian groups – additive and multiplicative. Generally, the notation is the usual notation for groups, while the additive notation is the usual notation for modules. To verify that a group is abelian, a table – known as a Cayley table – can be constructed in a similar fashion to a multiplication table. If the group is G = under the operation ⋅, the th entry of this contains the product gi ⋅ gj. The group is abelian if and only if this table is symmetric about the main diagonal and this is true since if the group is abelian, then gi ⋅ gj = gj ⋅ gi. This implies that the th entry of the table equals the th entry, every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. Thus the integers, Z, form a group under addition, as do the integers modulo n. Every ring is a group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group, in particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication
22.
Dihedral group
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, in abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article, a regular polygon with n sides has 2 n different symmetries, n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here. The associated rotations and reflections make up the dihedral group D n, if n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides, in either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes, as with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3
23.
Nilpotent group
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In group theory, a nilpotent group is a group that is almost abelian. This idea is motivated by the fact that nilpotent groups are solvable and it is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov, nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups, analogous terms are used for Lie algebras including nilpotent, lower central series, and upper central series. The definition uses the idea, explained on its own page, the following are equivalent formulations, A nilpotent group is one that has a central series of finite length. A nilpotent group is one whose lower central series terminates in the trivial subgroup after finitely many steps, a nilpotent group is one whose upper central series terminates in the whole group after finitely many steps. For a nilpotent group, the smallest n such that G has a series of length n is called the nilpotency class of G. Equivalently, the class of G equals the length of the lower central series or upper central series. If a group has nilpotency class at most m, then it is called a nil- m group. As noted above, every group is nilpotent. For a small example, consider the quaternion group Q8. It has center of order 2, and its upper central series is, Q8, all finite p-groups are in fact nilpotent. The maximal class of a group of order pn is n -1, the 2-groups of maximal class are the generalised quaternion groups, the dihedral groups, and the semidihedral groups. The direct product of two nilpotent groups is nilpotent, conversely, every finite nilpotent group is the direct product of p-groups. The Heisenberg group is an example of non-abelian, infinite nilpotent group, the multiplicative group of upper unitriangular n x n matrices over any field F is a nilpotent group of nilpotent length n -1. The multiplicative group of upper triangular n x n matrices over a field F is not in general nilpotent. This is not a characteristic of nilpotent groups, groups for which ad g is nilpotent of degree n are called n-Engel groups. They are proven to be nilpotent if they have finite order, an abelian group is precisely one for which the adjoint action is not just nilpotent but trivial
24.
Solvable group
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In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a group is a group whose derived series terminates in the trivial subgroup. Historically, the word solvable arose from Galois theory and the proof of the unsolvability of quintic equation. Specifically, an equation is solvable by radicals if and only if the corresponding Galois group is solvable. Or equivalently, if its derived series, the normal series G ▹ G ▹ G ▹ ⋯. These two definitions are equivalent, since for every group H and every normal subgroup N of H, the least n such that G = is called the derived length of the solvable group G. For finite groups, an equivalent definition is that a group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a group has finite composition length. The Jordan–Hölder theorem guarantees that if one composition series has this property, for the Galois group of a polynomial, these cyclic groups correspond to nth roots over some field. All abelian groups are trivially solvable – a subnormal series being given by just the group itself, but non-abelian groups may or may not be solvable. More generally, all nilpotent groups are solvable, in particular, finite p-groups are solvable, as all finite p-groups are nilpotent. A small example of a solvable, non-nilpotent group is the symmetric group S3, in fact, as the smallest simple non-abelian group is A5, it follows that every group with order less than 60 is solvable. The group S5 is not solvable — it has a series, giving factor groups isomorphic to A5 and C2. Generalizing this argument, coupled with the fact that An is a normal, maximal, non-abelian simple subgroup of Sn for n >4, we see that Sn is not solvable for n >4. This is a key step in the proof that for every n >4 there are polynomials of n which are not solvable by radicals. This property is used in complexity theory in the proof of Barringtons theorem. The celebrated Feit–Thompson theorem states that every group of odd order is solvable. In particular this implies that if a group is simple
25.
Classification of finite simple groups
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In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four broad classes described below. These groups can be seen as the building blocks of all finite groups. The Jordan–Hölder theorem is a precise way of stating this fact about finite groups. The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors. Gorenstein, Lyons, and Solomon are gradually publishing a simplified and revised version of the proof, the classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups can sometimes be reduced to questions about finite simple groups. Thanks to the theorem, such questions can sometimes be answered by checking each family of simple groups. Daniel Gorenstein announced in 1983 that the simple groups had all been classified. The completed proof of the classification was announced by Aschbacher after Aschbacher, the simple groups of small 2-rank include, Groups of 2-rank 0, in other words groups of odd order, which are all solvable by the Feit–Thompson theorem. Alperin showed that the Sylow subgroup must be dihedral, quasidihedral, wreathed, Groups of sectional 2-rank at most 4, classified by the Gorenstein–Harada theorem. All groups not of small 2 rank can be split into two classes, groups of component type and groups of characteristic 2 type. A group is said to be of component type if for some centralizer C of an involution and these are more or less the groups of Lie type of odd characteristic of large rank, and alternating groups, together with some sporadic groups. A major step in case is to eliminate the obstruction of the core of an involution. This is accomplished by the B-theorem, which states that every component of C/O is the image of a component of C. The idea is that groups have a centralizer of an involution with a component that is a smaller quasisimple group. So to classify these groups one takes every central extension of known finite simple group. A group is of characteristic 2 type if the generalized Fitting subgroup F* of every 2-local subgroup Y is a 2-group. As the name suggests these are roughly the groups of Lie type over fields of characteristic 2, the rank 1 groups are the thin groups, classified by Aschbacher, and the rank 2 ones are the notorious quasithin groups, classified by Aschbacher and Smith. These correspond roughly to groups of Lie type of ranks 1 or 2 over fields of characteristic 2
26.
Alternating group
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In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the group of degree n, or the alternating group on n letters. For n >1, the group An is the subgroup of the symmetric group Sn with index 2 and has therefore n. It is the kernel of the signature group homomorphism sgn. The group An is abelian if and only if n ≤3 and simple if, a5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group. The group A4 has a Klein four-group V as a normal subgroup, namely the identity and the double transpositions. As in the group, the conjugacy classes in An consist of elements with the same cycle shape. Examples, The two permutations and are not conjugates in A3, although they have the same cycle shape, the permutation is not conjugate to its inverse in A8, although the two permutations have the same cycle shape, so they are conjugate in S8. An is generated by 3-cycles, since 3-cycles can be obtained by combining pairs of transpositions and this generating set is often used to prove that An is simple for n ≥5. For n =1 and 2, the group is trivial. For n =3 the automorphism group is Z2, with trivial inner automorphism group, the outer automorphism group of A6 is the Klein four-group V = Z2 × Z2, and is related to the outer automorphism of S6. The extra outer automorphism in A6 swaps the 3-cycles with elements of shape 32, there are some exceptional isomorphisms between some of the small alternating groups and small groups of Lie type, particularly projective special linear groups. These are, A4 is isomorphic to PSL2 and the group of chiral tetrahedral symmetry. A5 is isomorphic to PSL2, PSL2, and the group of chiral icosahedral symmetry. A6 is isomorphic to PSL2 and PSp4, more obviously, A3 is isomorphic to the cyclic group Z3, and A0, A1, and A2 are isomorphic to the trivial group. A subgroup of three elements with any additional element generates the whole group, for all n ≠4, An has no nontrivial normal subgroups. Thus, An is a group for all n ≠4. A5 is the smallest non-solvable group, the group homology of the alternating groups exhibits stabilization, as in stable homotopy theory, for sufficiently large n, it is constant. However, there are some low-dimensional exceptional homology, note that the homology of the symmetric group exhibits similar stabilization, but without the low-dimensional exceptions
27.
Sporadic group
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In group theory, a discipline within mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the trivial group, the classification theorem states that the list of finite simple groups consists of 18 countably infinite families, plus 26 exceptions that do not follow such a systematic pattern. They are also known as the simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is regarded as a sporadic group. The monster group is the largest of the groups and contains all. Five of the groups were discovered by Mathieu in the 1860s. Several of these groups were predicted to exist before they were constructed, most of the groups are named after the mathematician who first predicted their existence. In some other sources, the Tits group is regarded as neither sporadic nor of Lie type, matrix representations over finite fields for all the sporadic groups have been constructed. The diagram on the right is based on the diagram given in Ronan, the sporadic groups also have a lot of subgroups which are not sporadic but these are not shown on the diagram because they are too numerous. Of the 26 sporadic groups,20 can be seen inside the Monster group as subgroups or quotients of subgroups, the six exceptions are J1, J3, J4, ON, Ru and Ly. These six are known as the pariahs. The remaining twenty have been called the Happy Family by Robert Griess, mn for n =11,12,22,23 and 24 are multiply transitive permutation groups on n points. They are all subgroups of M24, which is a group on 24 points. Finally, the Monster group itself is considered to be in this generation, the Tits group also belongs in this generation, there is a subgroup S4 ×2F4′ normalising a 2C2 subgroup of B, giving rise to a subgroup 2·S4 ×2F4′ normalising a certain Q8 subgroup of the Monster. 2F4′ is also a subgroup of the Fischer groups Fi22, Fi23 and Fi24′, 2F4′ is also a subgroup of the Rudvalis group Ru, and has no involvements in sporadic simple groups except the containments we have already mentioned. Atlas of Finite Group Representations, Sporadic groups
28.
Lagrange's theorem (group theory)
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Lagranges theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange and this can be shown using the concept of left cosets of H in G. If we can show that all cosets of H have the number of elements. This map is bijective because its inverse is given by f −1 = a b −1 y and this proof also shows that the quotient of the orders |G| / |H| is equal to the index. If we allow G and H to be infinite, and write this statement as | G | = ⋅ | H |, then, seen as a statement about cardinal numbers, it is equivalent to the axiom of choice. A consequence of the theorem is that the order of any element a of a group divides the order of that group. If the group has n elements, it follows a n = e and this can be used to prove Fermats little theorem and its generalization, Eulers theorem. These special cases were known long before the general theorem was proved, the theorem also shows that any group of prime order is cyclic and simple. This in turn can be used to prove Wilsons theorem, that if p is prime p is a factor of. Hence p < q, contradicting the assumption that p is the largest prime, Lagranges theorem raises the converse question as to whether every divisor of the order of a group is the order of some subgroup. This does not hold in general, given a finite group G, the smallest example is the alternating group G = A4, which has 12 elements but no subgroup of order 6. A CLT group is a group with the property that for every divisor of the order of the group. It is known that a CLT group must be solvable and that every group is a CLT group. There are partial converses to Lagranges theorem, for solvable groups, Halls theorems assert the existence of a subgroup of order equal to any unitary divisor of the group order. Lagrange did not prove Lagranges theorem in its general form, the number of such polynomials is the index in the symmetric group Sn of the subgroup H of permutations that preserve the polynomial. So the size of H divides n, with the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name. In his Disquisitiones Arithmeticae in 1801, Carl Friedrich Gauss proved Lagranges theorem for the case of Z*, the multiplicative group of nonzero integers modulo p. In 1844, Augustin-Louis Cauchy proved Lagranges theorem for the symmetric group Sn, camille Jordan finally proved Lagranges theorem for the case of any permutation group in 1861
29.
Sylow theorems
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The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime p, a Sylow p-subgroup of a group G is a maximal p-subgroup of G, i. e. a subgroup of G that is a p-group that is not a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a prime p is sometimes written Sylp. The Sylow theorems assert a partial converse to Lagranges theorem, the order of a Sylow p-subgroup of a finite group G is pn, where n is the multiplicity of p in the order of G, and every subgroup of order pn is a Sylow p-subgroup of G. The Sylow p-subgroups of a group are conjugate to each other, the number of Sylow p-subgroups of a group for a given prime p is congruent to 1 mod p. Collections of subgroups that are each maximal in one sense or another are common in group theory. That is, P is a p-group and gcd =1 and these properties can be exploited to further analyze the structure of G. The following theorems were first proposed and proven by Ludwig Sylow in 1872, Theorem 1, For every prime factor p with multiplicity n of the order of a finite group G, there exists a Sylow p-subgroup of G, of order pn. The following weaker version of theorem 1 was first proved by Cauchy, and is known as Cauchys theorem. Theorem 3, Let p be a factor with multiplicity n of the order of a finite group G, so that the order of G can be written as pnm. Let np be the number of Sylow p-subgroups of G, Then the following hold, np divides m, which is the index of the Sylow p-subgroup in G. np ≡1. Np = |G, NG|, where P is any Sylow p-subgroup of G, the Sylow theorems imply that for a prime number p every Sylow p-subgroup is of the same order, pn. Conversely, if a subgroup has order pn, then it is a Sylow p-subgroup, due to the maximality condition, if H is any p-subgroup of G, then H is a subgroup of a p-subgroup of order pn. A very important consequence of Theorem 2 is that the condition np =1 is equivalent to saying that the Sylow p-subgroup of G is a normal subgroup, there is an analogue of the Sylow theorems for infinite groups. We define a Sylow p-subgroup in a group to be a p-subgroup that is maximal for inclusion among all p-subgroups in the group. Such subgroups exist by Zorns lemma, Theorem, If K is a Sylow p-subgroup of G, and np = |Cl| is finite, then every Sylow p-subgroup is conjugate to K, and np ≡1, where Cl denotes the conjugacy class of K. A simple illustration of Sylow subgroups and the Sylow theorems are the group of the n-gon. For n odd,2 =21 is the highest power of 2 dividing the order, and thus subgroups of order 2 are Sylow subgroups. These are the groups generated by a reflection, of which there are n, and they are all conjugate under rotations, geometrically the axes of symmetry pass through a vertex and a side
30.
Frobenius group
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In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius, a subgroup H of a Frobenius group G fixing a point of the set X is called the Frobenius complement. The identity element together with all elements not in any conjugate of H form a subgroup called the Frobenius kernel K. The Frobenius group G is the product of K and H, G = K ⋊ H. Both the Frobenius kernel and the Frobenius complement have very restricted structures, J. G. Thompson proved that the Frobenius kernel K is a nilpotent group. If H has even order then K is abelian, the Frobenius complement H has the property that every subgroup whose order is the product of 2 primes is cyclic, this implies that its Sylow subgroups are cyclic or generalized quaternion groups. Any group such that all Sylow subgroups are cyclic is called a Z-group, and in particular must be a metacyclic group, this means it is the extension of two cyclic groups. If a Frobenius complement H is not solvable then Zassenhaus showed that it has a subgroup of index 1 or 2 that is the product of SL2. In particular, if a Frobenius complement coincides with its derived subgroup, if a Frobenius complement H is solvable then it has a normal metacyclic subgroup such that the quotient is a subgroup of the symmetric group on 4 points. The Frobenius kernel K is uniquely determined by G as it is the Fitting subgroup, in particular a finite group G is a Frobenius group in at most one way. The smallest example is the group on 3 points, with 6 elements. The Frobenius kernel K has order 3, and the complement H has order 2, for every finite field Fq with q elements, the group of invertible affine transformations x ↦ a x + b, a ≠0 acting naturally on Fq is a Frobenius group. The preceding example corresponds to the case F3, the field with three elements, identifying F8* with the Fano plane, σ can be taken to be the restriction of the Frobenius automorphism σ=x² of F8 and τ to be multiplication by any element not in the prime field F2. This Frobenius group acts transitively on the 21 flags in the Fano plane. The dihedral group of order 2n with n odd is a Frobenius group with complement of order 2. More generally if K is any group of odd order and H has order 2 and acts on K by inversion. Many further examples can be generated by the following constructions, if we replace the Frobenius complement of a Frobenius group by a non-trivial subgroup we get another Frobenius group. If we have two Frobenius groups K1. H and K2. H then. H is also a Frobenius group
31.
Schur multiplier
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In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H2 of a group G. It was introduced by Issai Schur in his work on projective representations, the Schur multiplier M of a finite group G is a finite abelian group whose exponent divides the order of G. If a Sylow p-subgroup of G is cyclic for some p, in particular, if all Sylow p-subgroups of G are cyclic, then M is trivial. For instance, the Schur multiplier of the group of order 6 is the trivial group since every Sylow subgroup is cyclic. The Schur multiplier of the abelian group of order 16 is an elementary abelian group of order 64. The Schur multiplier of the group is trivial, but the Schur multiplier of dihedral 2-groups has order 2. The Schur multipliers of the simple groups are given at the list of finite simple groups. The covering groups of the alternating and symmetric groups are of recent interest. Schurs original motivation for studying the multiplier was to classify projective representations of a group, in other words, a projective representation is a representation modulo the center. The Schur cover is known as a covering group or Darstellungsgruppe. The Schur covers of the simple groups are known. The Schur cover of a group is uniquely determined up to isomorphism. The study of such covering groups led naturally to the study of central, a central extension of a group G is an extension 1 → K → C → G →1 where K ≤ Z is a subgroup of the center of C. If the finite group G is moreover perfect, then C is unique up to isomorphism and is itself perfect, such C are often called universal perfect central extensions of G, or covering group. If the finite group G is not perfect, then its Schur covering groups are only isoclinic, stem extensions have the nice property that any lift of a generating set of G is a generating set of C. Since the relations of G specify elements of K when considered as part of C, in fact if G is perfect, this is all that is needed, C ≅ / and M ≅ K ≅ R/. Because of this simplicity, expositions such as handle the case first. The general case for the Schur multiplier is similar but ensures the extension is an extension by restricting to the derived subgroup of F, M ≅ /
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Symmetric group
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Since there are n. possible permutation operations that can be performed on a tuple composed of n symbols, it follows that the order of the symmetric group Sn is n. For the remainder of this article, symmetric group will mean a group on a finite set. The symmetric group is important to diverse areas of such as Galois theory, invariant theory, the representation theory of Lie groups. Cayleys theorem states that every group G is isomorphic to a subgroup of the group on G. The symmetric group on a finite set X is the group elements are all bijective functions from X to X. For finite sets, permutations and bijective functions refer to the same operation, the symmetric group of degree n is the symmetric group on the set X =. The symmetric group on a set X is denoted in various ways including SX,
33.
Klein four-group
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In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe by Felix Klein in 1884, with four elements, the Klein four-group is the smallest non-cyclic group, and the cyclic group of order 4 and the Klein four-group are, up to isomorphism, the only groups of order 4. The smallest non-abelian group is the group of degree 3. The Klein groups Cayley table is given by, The Klein four-group is also defined by the group presentation V = ⟨ a, b ∣ a 2 = b 2 =2 = e ⟩. All non-identity elements of the Klein group have order 2, thus any two non-identity elements can serve as generators in the above presentation, the Klein four-group is the smallest non-cyclic group. It is however a group, and isomorphic to the dihedral group of order 4, Dih2, other than the group of order 2. The Klein four-group is also isomorphic to the direct sum Z2 ⊕ Z2, so that it can be represented as the pairs under component-wise addition modulo 2, the Klein four-group is thus an example of an elementary abelian 2-group, which is also called a Boolean group. Another numerical construction of the Klein four-group is the set, with the operation being multiplication modulo 8, here a is 3, b is 5, and c = ab is 3 ×5 =15 ≡7. The three elements of two in the Klein four-group are interchangeable, the automorphism group of V is the group of permutations of these three elements. In fact, it is the kernel of a group homomorphism from S4 to S3. In the construction of finite rings, eight of the rings with four elements have the Klein four-group as their additive substructure. The quotient group / is isomorphic to the Klein four-group, in a similar fashion, the group of units of the split-complex number ring, when divided by its identity component, also results in the Klein four-group. The Klein four-group as a subgroup of the alternating group A4 is not the group of any simple graph. It is, however, the group of a two-vertex graph where the vertices are connected to each other with two edges, making the graph non-simple. A. Armstrong Groups and Symmetry, Springer Verlag, page 53, W. E. Barnes Introduction to Abstract Algebra, D. C
34.
Quaternion group
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In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication. The Cayley table for Q is given by, The quaternion group has the property of being Hamiltonian, every subgroup of Q is a normal subgroup. Every Hamiltonian group contains a copy of Q, the quaternion group is the smallest example of nilpotent non-abelian group. Another example of nilpotent non-abelian group is group of order eight. The quaternion group has five irreducible representations, and their dimensions are 1,1,1,1,2, respectively. The proof for this property is not hard, since the number of characters of the quaternion group is equal to the number of conjugacy classes of the quaternion group. These five representations are, Trivial representation Sign representations with i, j, k-kernel, for each maximal normal subgroup, we obtain a one-dimensional representation with that subgroup as kernel. The representation sends elements inside the subgroup to 1, and elements outside the subgroup to -1, 2-dimensional representation, A representation, Q = → G L2 stated in Matrix representations. So the character table of the group is, In abstract algebra. The result is a field called the quaternions. Note that this is not quite the same as the group algebra on Q, conversely, one can start with the quaternions and define the quaternion group as the multiplicative subgroup consisting of the eight elements. The complex four-dimensional vector space on the basis is called the algebra of biquaternions. Note that i, j, and k all have four in Q. Another presentation of Q demonstrating this is, ⟨ x, y ∣ x 4 =1, x 2 = y 2, y −1 x y = x −1 ⟩. One may take, for instance, i = x, j = y and k = xy, the center and the commutator subgroup of Q is the subgroup. The factor group Q/ is isomorphic to the Klein four-group V, the inner automorphism group of Q is isomorphic to Q modulo its center, and is therefore also isomorphic to the Klein four-group. The full automorphism group of Q is isomorphic to the group of degree 4, S4. The outer automorphism group of Q is then S4/V which is isomorphic to S3, the quaternion group can be represented as a subgroup of the general linear group GL2
35.
Dicyclic group
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In group theory, a dicyclic group is a member of a class of non-abelian groups of order 4n. It is an extension of the group of order 2 by a cyclic group of order 2n. In the notation of exact sequences of groups, this extension can be expressed as,1 → C2 n → Dic n → C2 →1, more generally, given any finite abelian group with an order-2 element, one can define a dicyclic group. Some things to note which follow from this definition, x4 =1 x2ak = ak+n = akx2 if j = ±1, thus, every element of Dicn can be uniquely written as akxj, where 0 ≤ k < 2n and j =0 or 1. When n =2, the group is isomorphic to the quaternion group Q. More generally, when n is a power of 2, the group is isomorphic to the generalized quaternion group. For each n >1, the dicyclic group Dicn is a group of order 4n. Let A = <a> be the subgroup of Dicn generated by a, then A is a cyclic group of order 2n, so =2. As a subgroup of index 2 it is automatically a normal subgroup, the quotient group Dicn/A is a cyclic group of order 2. Dicn is solvable, note that A is normal, and being abelian, is itself solvable, there is a superficial resemblance between the dicyclic groups and dihedral groups, both are a sort of mirroring of an underlying cyclic group. But the presentation of a group would have x2 =1, instead of x2 = an. In particular, Dicn is not a product of A and <x>. The dicyclic group has an involution, namely x2 = an. Note that this element lies in the center of Dicn, indeed, the center consists solely of the identity element and x2. If we add the relation x2 =1 to the presentation of Dicn one obtains a presentation of the dihedral group Dih2n, there is a natural 2-to-1 homomorphism from the group of unit quaternions to the 3-dimensional rotation group described at quaternions and spatial rotations. Since the dicyclic group can be embedded inside the unit quaternions one can ask what the image of it is under this homomorphism, the answer is just the dihedral symmetry group Dihn. For this reason the group is also known as the binary dihedral group. Note that the group does not contain any subgroup isomorphic to Dihn
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Discrete group
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For example, the integers, Z, form a discrete subgroup of the reals, R, but the rational numbers, Q, do not. A discrete group is a topological group G equipped with the discrete topology, any group can be given the discrete topology. Since every map from a space is continuous, the topological homomorphisms between discrete groups are exactly the group homomorphisms between the underlying groups. Hence, there is an isomorphism between the category of groups and the category of discrete groups, discrete groups can therefore be identified with their underlying groups. There are some occasions when a group or Lie group is usefully endowed with the discrete topology. This happens for example in the theory of the Bohr compactification, a discrete isometry group is an isometry group such that for every point of the metric space the set of images of the point under the isometries is a discrete set. A discrete symmetry group is a group that is a discrete isometry group. Since topological groups are homogeneous, one need look at a single point to determine if the topological group is discrete. In particular, a group is discrete if and only if the singleton containing the identity is an open set. A discrete group is the thing as a zero-dimensional Lie group. The identity component of a group is just the trivial subgroup while the group of components is isomorphic to the group itself. Since the only Hausdorff topology on a set is the discrete one. It follows that every subgroup of a Hausdorff group is discrete. A discrete normal subgroup of a connected group G necessarily lies in the center of G and is therefore abelian, other properties, every discrete group is totally disconnected every subgroup of a discrete group is discrete. Every quotient of a group is discrete. The product of a number of discrete groups is discrete. A discrete group is compact if and only if it is finite, every discrete group is locally compact. Every discrete subgroup of a Hausdorff group is closed, every discrete subgroup of a compact Hausdorff group is finite
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Lattice (discrete subgroup)
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In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. The theory is particularly rich for lattices in semisimple Lie groups or more generally in semisimple algebraic groups over local fields. In particular there is a wealth of rigidity results in this setting, lattices are also well-studied in some other classes of groups, in particular groups associated to Kac-Moody algebras and automorphisms groups of regular trees. Lattices are of interest in areas of mathematics, geometric group theory, in differential geometry, in number theory, in ergodic theory. Lattices are best thought of as discrete approximations of continuous groups, maybe the most obvious idea is to say that a subgroup approximates a larger group is that the larger group can be covered by the translates of a small subset by all elements in the subgroups. In a locally compact topological group there are two immediately available notions of small, topological or measure-theoretical, note that since the Haar measure is a Borel measure, in particular gives finite mass to compact subsets, the second definition is more inclusive. The definition of a used in mathematics relies upon the second meaning. Let G be a compact group and Γ a discrete subgroup. Then Γ is called a lattice in G if in addition there exists a Borel measure μ on the quotient space G / Γ which is finite, then Γ is a lattice if and only if this measure is finite. A lattice Γ ⊂ G is called uniform when the quotient space G / Γ is compact, equivalently a discrete subgroup Γ ⊂ G is a uniform lattice if and only if there exists a compact subset C ⊂ G with G = ⋃ γ ∈ Γ C γ. Note that if Γ is any discrete subgroup in G such that G / Γ is compact then Γ is automatically a lattice in G, the fundamental, and simplest, example is the subgroup Z n which is a lattice in the Lie group R n. A slightly more complicated example is given by the discrete Heisenberg group inside the continuous Heisenberg group, If G is a discrete group then a lattice in G is exactly a subgroup Γ of finite index. All of these examples are uniform, a non-uniform example is given by the modular group S L2 inside S L2, and also by the higher-dimensional analogues S L n ⊂ S L n. Any finite-index subgroup of a lattice is also a lattice in the same group, more generally, a subgroup commensurable to a lattice is a lattice. Not every locally compact group contains a lattice, and there is no general group-theoretical sufficient condition for this, on the other hand, there are plenty of more specific settings where such criteria exist. For example, the existence or non-existence of lattices in Lie groups is a well-understood topic, as we mentioned, a necessary condition for a group to contain a lattice is that it be unimodular. This allows for the construction of groups without lattices, for example the group of invertible upper triangular matrices or the affine groups. It is also not very hard to find unimodular groups without lattices, a stronger condition than unimodularity is simplicity
38.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
39.
Free group
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The members of S are called generators of FS. A related but different notion is an abelian group, both notions are particular instances of a free object from universal algebra. Free groups first arose in the study of geometry, as examples of Fuchsian groups. In an 1882 paper, Walther von Dyck pointed out that groups have the simplest possible presentations. The algebraic study of groups was initiated by Jakob Nielsen in 1924. Max Dehn realized the connection topology, and obtained the first proof of the full Nielsen–Schreier theorem. Otto Schreier published a proof of this result in 1927. Later on in the 1930s, Wilhelm Magnus discovered the connection between the central series of free groups and free Lie algebras. The group of integers is free, we can take S =, a free group on a two-element set S occurs in the proof of the Banach–Tarski paradox and is described there. On the other hand, any finite group cannot be free. In algebraic topology, the group of a bouquet of k circles is the free group on a set of k elements. The free group FS with free generating set S can be constructed as follows, S is a set of symbols, and we suppose for every s in S there is a corresponding inverse symbol, s−1, in a set S−1. Let T = S ∪ S−1, and define a word in S to be any written product of elements of T and that is, a word in S is an element of the monoid generated by T. The empty word is the word with no symbols at all, for example, if S =, then T =, and a b 3 c −1 c a −1 c is a word in S. If an element of S lies immediately next to its inverse, the word may be simplified by omitting the c, c−1 pair, a word that cannot be simplified further is called reduced. The free group FS is defined to be the group of all reduced words in S, the identity is the empty word. A word is called cyclically reduced, if its first and last letter are not inverse to each other, Every word is conjugate to a cyclically reduced word, and a cyclically reduced conjugate of a cyclically reduced word is a cyclic permutation of the letters in the word. For instance b−1abcb is not cyclically reduced, but is conjugate to abc, the only cyclically reduced conjugates of abc are abc, bca, and cab
40.
Modular group
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In mathematics, the modular group is the projective special linear group PSL of 2 x 2 matrices with integer coefficients and unit determinant. The matrices A and -A are identified, the group operation is function composition. This group of transformations is isomorphic to the special linear group PSL. In other words, PSL consists of all matrices where a, b, c, and d are integers, ad − bc =1, the group operation is the usual multiplication of matrices. Some authors define the group to be PSL, and still others define the modular group to be the larger group SL. Some mathematical relations require the consideration of the group GL of matrices with determinant plus or minus one, similarly, PGL is the quotient group GL/. A2 ×2 matrix with unit determinant is a matrix, and thus SL = Sp. The unit determinant of implies that the fractions a/b, a/c, c/d and b/d are all irreducible, more generally, if p/q is an irreducible fraction, then a p + b q c p + d q is also irreducible. Elements of the group provide a symmetry on the two-dimensional lattice. Let ω1 and ω2 be two numbers whose ratio is not real. Then the set of points Λ = is a lattice of parallelograms on the plane, a different pair of vectors α1 and α2 will generate exactly the same lattice if and only if = for some matrix in GL. It is for this reason that doubly periodic functions, such as elliptic functions, the action of the modular group on the rational numbers can most easily be understood by envisioning a square grid, with grid point corresponding to the fraction p/q. An irreducible fraction is one that is visible from the origin, the action of the group on a fraction never takes a visible to a hidden one. If p n −1 / q n −1 and p n / q n are two successive convergents of a fraction, then the matrix belongs to GL. In particular, if bc − ad =1 for positive integers a, b, c and d with a < b and c < d then a/b, important special cases of continued fraction convergents include the Fibonacci numbers and solutions to Pells equation. In both cases, the numbers can be arranged to form a subset of the modular group. Geometrically, S represents inversion in the unit followed by reflection with respect to the imaginary axis. The generators S and T obey the relations S2 =1 and 3 =1