1.
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
2.
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
3.
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
4.
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
5.
Lie group
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In mathematics, a Lie group /ˈliː/ is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie’s student Arthur Tresse, an extension of Galois theory to the case of continuous symmetry groups was one of Lies principal motivations. Lie groups are smooth manifolds and as such can be studied using differential calculus. Lie groups play an role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various geometries by specifying an appropriate transformation group that leaves certain geometric properties invariant and this idea later led to the notion of a G-structure, where G is a Lie group of local symmetries of a manifold. On a global level, whenever a Lie group acts on an object, such as a Riemannian or a symplectic manifold. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry, Linear actions of Lie groups are especially important, and are studied in representation theory. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, a real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication μ, G × G → G μ = x y means that μ is a mapping of the product manifold G×G into G. These two requirements can be combined to the requirement that the mapping ↦ x −1 y be a smooth mapping of the product manifold into G. The 2×2 real invertible matrices form a group under multiplication, denoted by GL or by GL2 and this is a four-dimensional noncompact real Lie group. This group is disconnected, it has two connected components corresponding to the positive and negative values of the determinant, the rotation matrices form a subgroup of GL, denoted by SO. It is a Lie group in its own right, specifically, using the rotation angle φ as a parameter, this group can be parametrized as follows, SO =. Addition of the angles corresponds to multiplication of the elements of SO, thus both multiplication and inversion are differentiable maps. The orthogonal group also forms an example of a Lie group. All of the examples of Lie groups fall within the class of classical groups. Hilberts fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples, if the underlying manifold is allowed to be infinite-dimensional, then one arrives at the notion of an infinite-dimensional Lie group
6.
Solenoid (mathematics)
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This page discusses a class of topological groups. For the wrapped loop of wire, see Solenoid and this construction can be carried out geometrically in the three-dimensional Euclidean space R3. A solenoid is a one-dimensional homogeneous indecomposable continuum that has the structure of a topological group. Such a solenoid arises as an expanding attractor, or Smale–Williams attractor. Each solenoid may be constructed as the intersection of a system of embedded solid tori in R3. Fix a sequence of numbers, ni ≥2. Let T0 = S1 × D be a solid torus, for each i ≥0, choose a solid torus Ti+1 that is wrapped longitudinally ni times inside the solid torus Ti. Then their intersection Λ = ⋂ i ≥0 T i is homeomorphic to the solenoid constructed as the limit of the system of circles with the maps determined by the sequence. Here is a variant of this construction isolated by Stephen Smale as an example of an attractor in the theory of smooth dynamical systems. Denote the angular coordinate on the circle S1 by t and consider the coordinate z on the two-dimensional unit disk D. Let f be the map of the solid torus T = S1 × D into itself given by the formula f =. This map is an embedding of T into itself that preserves the foliation by meridional disks. Solenoids are compact metrizable spaces that are connected, but not locally connected or path connected and this is reflected in their pathological behavior with respect to various homology theories, in contrast with the standard properties of homology for simplicial complexes. In Čech homology, one can construct a non-exact long homology sequence using a solenoid, in Steenrod-style homology theories, the 0th homology group of a solenoid may have a fairly complicated structure, even though a solenoid is a connected space. Protorus, a class of groups that includes the solenoids Pontryagin duality D. van Dantzig, Ueber topologisch homogene Kontinua. 15, pp. 102–125 Hazewinkel, Michiel, ed. Smale, Differentiable dynamical systems, L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. 97, pp. 454–472 Robert F. Williams, Expanding attractors,43, p. 169–203 Semmes, Stephen, Some remarks about solenoids, arXiv,1201.2647
7.
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
8.
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
9.
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
10.
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
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.
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