1.
Group order
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In group theory, a branch of mathematics, the term order is used in two unrelated senses, The order of a group is its cardinality, i. e. the number of elements in its set. Also, the order, sometimes period, of an element a of a group is the smallest positive integer m such that am = e, if no such m exists, a is said to have infinite order. The ordering relation of a partially or totally ordered group and this article is about the first sense of order. The order of a group G is denoted by ord or | G |, the symmetric group S3 has the following multiplication table. This group has six elements, so ord =6, by definition, the order of the identity, e, is 1. Each of s, t, and w squares to e, completing the enumeration, both u and v have order 3, for u2 = v and u3 = vu = e, and v2 = u and v3 = uv = e. The order of a group and that of an element tend to speak about the structure of the group, roughly speaking, the more complicated the factorization of the order the more complicated the group. If the order of group G is 1, then the group is called a trivial group, given an element a, ord =1 if and only if a is the identity. If every element in G is the same as its inverse, then ord =2 and consequently G is abelian since a b = −1 = b −1 a −1 = b a by Elementary group theory. The converse of this statement is not true, for example, the cyclic group Z6 of integers modulo 6 is abelian, but the number 2 has order 3,2 +2 +2 =6 ≡0. The relationship between the two concepts of order is the following, if we write ⟨ a ⟩ = for the subgroup generated by a, for any integer k, we have ak = e if and only if ord divides k. In general, the order of any subgroup of G divides the order of G, more precisely, if H is a subgroup of G, then ord / ord =, where is called the index of H in G, an integer. As an immediate consequence of the above, we see that the order of every element of a group divides the order of the group. For example, in the symmetric group shown above, where ord =6, the following partial converse is true for finite groups, if d divides the order of a group G and d is a prime number, then there exists an element of order d in G. The statement does not hold for composite orders, e. g. the Klein four-group does not have an element of order four) and this can be shown by inductive proof. The consequences of the include, the order of a group G is a power of a prime p if. If a has order, then all powers of a have infinite order as well. If a has order, we have the following formula for the order of the powers of a
2.
Order of a group element
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In group theory, a branch of mathematics, the term order is used in two unrelated senses, The order of a group is its cardinality, i. e. the number of elements in its set. Also, the order, sometimes period, of an element a of a group is the smallest positive integer m such that am = e, if no such m exists, a is said to have infinite order. The ordering relation of a partially or totally ordered group and this article is about the first sense of order. The order of a group G is denoted by ord or | G |, the symmetric group S3 has the following multiplication table. This group has six elements, so ord =6, by definition, the order of the identity, e, is 1. Each of s, t, and w squares to e, completing the enumeration, both u and v have order 3, for u2 = v and u3 = vu = e, and v2 = u and v3 = uv = e. The order of a group and that of an element tend to speak about the structure of the group, roughly speaking, the more complicated the factorization of the order the more complicated the group. If the order of group G is 1, then the group is called a trivial group, given an element a, ord =1 if and only if a is the identity. If every element in G is the same as its inverse, then ord =2 and consequently G is abelian since a b = −1 = b −1 a −1 = b a by Elementary group theory. The converse of this statement is not true, for example, the cyclic group Z6 of integers modulo 6 is abelian, but the number 2 has order 3,2 +2 +2 =6 ≡0. The relationship between the two concepts of order is the following, if we write ⟨ a ⟩ = for the subgroup generated by a, for any integer k, we have ak = e if and only if ord divides k. In general, the order of any subgroup of G divides the order of G, more precisely, if H is a subgroup of G, then ord / ord =, where is called the index of H in G, an integer. As an immediate consequence of the above, we see that the order of every element of a group divides the order of the group. For example, in the symmetric group shown above, where ord =6, the following partial converse is true for finite groups, if d divides the order of a group G and d is a prime number, then there exists an element of order d in G. The statement does not hold for composite orders, e. g. the Klein four-group does not have an element of order four) and this can be shown by inductive proof. The consequences of the include, the order of a group G is a power of a prime p if. If a has order, then all powers of a have infinite order as well. If a has order, we have the following formula for the order of the powers of a
3.
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
4.
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
5.
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
6.
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 =
7.
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