1.
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
2.
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
3.
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
4.
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 =
5.
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
6.
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
7.
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
8.
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