Direct product of groups
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 grouptheoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.
In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G ⊕ H. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.
Definition
Given groups G and H, the direct product G × H is defined as follows:
 The elements of G × H are ordered pairs (g, h), where g ∈ G and h ∈ H. That is, the set of elements of G × H is the Cartesian product of the sets G and H.
 The binary operation on G × H is defined componentwise:
(g_{1}, h_{1}) · (g_{2}, h_{2}) = (g_{1} · g_{2}, h_{1} · h_{2})
The resulting algebraic object satisfies the axioms for a group. Specifically:
 Associativity
 The binary operation on G × H is indeed associative.
 Identity
 The direct product has an identity element, namely (1_{G}, 1_{H}), where 1_{G} is the identity element of G and 1_{H} is the identity element of H.
 Inverses
 The inverse of an element (g, h) of G × H is the pair (g^{−1}, h^{−1}), where g^{−1} is the inverse of g in G, and h^{−1} is the inverse of h in H.
Examples
 Let R be the group of real numbers under addition. Then the direct product R × R is the group of all twocomponent vectors (x, y) under the operation of vector addition:
(x_{1}, y_{1}) + (x_{2}, y_{2}) = (x_{1} + x_{2}, y_{1} + y_{2}).
 Let G and H be cyclic groups with two elements each:


 Then the direct product G × H is isomorphic to the Klein fourgroup:
G × H * (1, 1) (a, 1) (1, b) (a, b) (1, 1) (1, 1) (a, 1) (1, b) (a, b) (a, 1) (a, 1) (1, 1) (a, b) (1, b) (1, b) (1, b) (a, b) (1, 1) (a, 1) (a, b) (a, b) (1, b) (a, 1) (1, 1)
Elementary properties
 The order of a direct product G × H is the product of the orders of G and H:
  G × H  =  G   H .
 This follows from the formula for the cardinality of the cartesian product of sets.
 The order of each element (g, h) is the least common multiple of the orders of g and h:^{[1]}
  (g, h)  = lcm(  g ,  h  ).
 In particular, if  g  and  h  are relatively prime, then the order of (g, h) is the product of the orders of g and h .
 As a consequence, if G and H are cyclic groups whose orders are relatively prime, then G × H is cyclic as well. That is, if m and n are relatively prime, then
 ( Z / mZ ) × ( Z / nZ ) ≅ Z / mnZ.
 This fact is closely related to the Chinese remainder theorem.
Algebraic structure
Let G and H be groups, let P = G × H, and consider the following two subsets of P:
 G′ = { (g, 1) : g ∈ G } and H′ = { (1, h) : h ∈ H }
Both of these are in fact subgroups of P, the first being isomorphic to G, and the second being isomorphic to H. 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.
These subgroups of P have the following three important properties: (Saying again that we identify G′ and H′ with G and H, respectively.)
 The intersection G ∩ H is trivial.
 Every element of P can be expressed as the product of an element of G and an element of H.
 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, then P is necessarily isomorphic to the direct product of G and H. In this situation, P is sometimes referred to as the internal direct product of its subgroups G and H.
In some contexts, the third property above is replaced by the following:
 3′. Both G and H are normal in P.
This property is equivalent to property 3, since the elements of two normal subgroups with trivial intersection necessarily commute, a fact which can be deduced by considering the commutator [g,h] of any g in G, h in H.
Examples
 Let V be the Klein fourgroup:
V * 1 a b c 1 1 a b c a a 1 c b b b c 1 a c c b a 1
 Then V is the internal direct product of the twoelement subgroups { 1, a } and { 1, b }.
 Let 〈a〉 be a cyclic group of order mn, where m and n are relatively prime. Then 〈a^{n}〉 and 〈a^{m}〉 are cyclic subgroups of orders m and n, respectively, and 〈a〉 is the internal direct product of these subgroups.
 Let C^{×} be the group of nonzero complex numbers under multiplication. Then C^{×} is the internal direct product of the circle group T of unit complex numbers and the group R^{+} of positive real numbers under multiplication.
 If n is odd, then the general linear group GL(n, R) is the internal direct product of the special linear group SL(n, R) and the subgroup consisting of all scalar matrices.
 Similarly, when n is odd the orthogonal group O(n, R) is the internal direct product of the special orthogonal group SO(n, R) and the twoelement subgroup { −I, I }, where I denotes the identity matrix.
 The symmetry group of a cube is the internal direct product of the subgroup of rotations and the twoelement group { −I, I }, where I is the identity element and −I is the point reflection through the center of the cube. A similar fact holds true for the symmetry group of an icosahedron.
 Let n be odd, and let D_{4n} be the dihedral group of order 4n:
 D_{4n} = 〈 r, s  r^{2n} = s^{2} = 1, sr = r^{−1}s 〉.
 Then D_{4n} is the internal direct product of the subgroup 〈 r^{2}, s 〉 (which is isomorphic to D_{2n}) and the twoelement subgroup { 1, r^{n} }.
Presentations
The algebraic structure of G × H can be used to give a presentation for the direct product in terms of the presentations of G and H. Specifically, suppose that
 G = 〈 S_{G}  R_{G} 〉 and H = 〈 S_{H}  R_{H} 〉,
where S_{G} and S_{H} are (disjoint) generating sets and R_{G} and R_{H} are defining relations. Then
 G × H = 〈 S_{G} ∪ S_{H}  R_{G} ∪ R_{H} ∪ R_{P} 〉
where R_{P} is a set of relations specifying that each element of S_{G} commutes with each element of S_{H}
For example, suppose that
 G = 〈 a  a^{3} = 1 〉 and H = 〈 b  b^{5} = 1 〉.
Then
 G × H = 〈 a, b  a^{3} = 1, b^{5} = 1, ab = ba 〉.
Normal structure
As mentioned above, the subgroups G and H are normal in G × H. Specifically, define functions π_{G}: G × H → G and π_{H}: G × H → H by
 π_{G}(g, h) = g and π_{H}(g, h) = h.
Then π_{G} and π_{H} are homomorphisms, known as projection homomorphisms, whose kernels are H and G, respectively.
It follows that G × H is an extension of G by H (or vice versa). In the case where G × H is a finite group, it follows that the composition factors of G × H are precisely the union of the composition factors of G and the composition factors of H.
Further properties
Universal property
{{#invoke:mainmain}} The direct product G × H can be characterized by the following universal property. Let π_{G}: G × H → G and π_{H}: G × H → H be the projection homomorphisms. Then for any group P and any homomorphisms ƒ_{G}: P → G and ƒ_{H}: P → H, there exists a unique homomorphism ƒ: P → G × H making the following diagram commute:
Specifically, the homomorphism ƒ is given by the formula
 ƒ(p) = ( ƒ_{G}(p), ƒ_{H}(p) ).
This is a special case of the universal property for products in category theory.
Subgroups
If A is a subgroup of G and B is a subgroup of H, then the direct product A × B is a subgroup of G × H. For example, the isomorphic copy of G in G × H is the product G × {1}, where {1} is the trivial subgroup of H.
If A and B are normal, then A × B is a normal subgroup of G × H. Moreover, the quotient (G × H) / (A × B) is isomorphic to the direct product of the quotients G / A and H / B:
 (G × H) / (A × B) ≅ (G / A) × (H / B).
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 = { (g, g) : g ∈ G }
which is not the direct product of two subgroups of G. Other subgroups include fiber products of G and H (see below). The subgroups of direct products are described by Goursat's lemma.
Conjugacy and centralizers
Two elements (g_{1}, h_{1}) and (g_{2}, h_{2}) are conjugate in G × H if and only if g_{1} and g_{2} are conjugate in G and h_{1} and h_{2} are conjugate in H. It follows that each conjugacy class in G × H is simply the Cartesian product of a conjugacy class in G and a conjugacy class in H.
Along the same lines, if (g, h) ∈ G × H, the centralizer of (g, h) is simply the product of the centralizers of g and h:
 C_{G×H}(g, h) = C_{G}(g) × C_{H}(h).
Similarly, the center of G × H is the product of the centers of G and :
 Z(G × H) = Z(G) × Z(H).
Normalizers behave in a more complex manner since not all subgroups of direct products themselves decompose as direct products.
Automorphisms and endomorphisms
If α is an automorphism of G and β is an automorphism of H, then the product function α × β: G × H → G × H defined by
 (α × β)(g, h) = (α(g), β(h))
is an automorphism of G × H. It follows that Aut(G × H) has a subgroup isomorphic to the direct product Aut(G) × Aut(H).
It is not true in general that every automorphism of G × H has the above form. (That is, Aut(G) × Aut(H) is often a proper subgroup of Aut(G × H).) For example, if G is any group, then there exists an automorphism σ of G × G that switches the two factors, i.e.
 σ(g_{1}, g_{2}) = (g_{2}, g_{1}).
For another example, the automorphism group of Z × Z is GL(2, Z), the group of all 2 × 2 matrices with integer entries and determinant ±1. This automorphism group is infinite, but only finitely many of the automorphisms have the form given above.
In general, every endomorphism of G × H can be written as a 2 × 2 matrix
where α is an endomorphism of G, δ is an endomorphism of H, and β: H → G and γ: G → H are homomorphisms. Such a matrix must have the property that every element in the image of α commutes with every element in the image of β, and every element in the image of γ commutes with every element in the image of δ.
When G and H are indecomposable, centerless groups, then the automorphism group is relatively straightforward, being Aut(G) × Aut(H) if G and H are not isomorphic, and Aut(G) wr 2 if G ≅ H, wr denotes the wreath product. This is part of the Krull–Schmidt theorem, and holds more generally for finite direct products.
Generalizations
Finite direct products
It is possible to take the direct product of more than two groups at once. Given a finite sequence G_{1}, ..., G_{n} of groups, the direct product
is defined as follows:
 The elements of G_{1} × ··· × G_{n} are tuples (g_{1}, ..., g_{n}), where g_{i} ∈ G_{i} for each i.
 The operation on G_{1} × ··· × G_{n} is defined componentwise:
(g_{1}, ..., g_{n})(g_{1}′, ..., g_{n}′) = (g_{1}g_{1}′, ..., g_{n}g_{n}′).
This has many of the same properties as the direct product of two groups, and can be characterized algebraically in a similar way.
Infinite direct products
It is also possible to take the direct product of an infinite number of groups. For an infinite sequence G_{1}, G_{2}, ... of groups, this can be defined just like the finite direct product of above, with elements of the infinite direct product being infinite tuples.
More generally, given an indexed family { G_{i} }_{i∈I} of groups, the direct product ∏_{i∈I} G_{i} is defined as follows:
 The elements of ∏_{i∈I} G_{i} are the elements of the infinite Cartesian product of the sets G_{i}, i.e. functions ƒ: I → U_{i∈I} G_{i} with the property that ƒ(i) ∈ G_{i} for each i.
 The product of two elements ƒ, g is defined componentwise:
(ƒ • g)(i) = ƒ(i) • g(i).
Unlike a finite direct product, the infinite direct product ∏_{i∈I} G_{i} is not generated by the elements of the isomorphic subgroups { G_{i} }_{i∈I}. Instead, these subgroups generate a subgroup of the direct product known as the infinite direct sum, which consists of all elements that have only finitely many nonidentity components.
Other products
Semidirect products
{{#invoke:mainmain}} Recall that a group P with subgroups G and H is isomorphic to the direct product of G and H as long as it satisfies the following three conditions:
 The intersection G ∩ H is trivial.
 Every element of P can be expressed as the product of an element of G and an element of H.
 Both G and H are normal in P.
A semidirect product of G and H is obtained by relaxing the third condition, so that only one of the two subgroups G, H is required to be normal. The resulting product still consists of ordered pairs (g, h), but with a slightly more complicated rule for multiplication.
It is also possible to relax the third condition entirely, requiring neither of the two subgroups to be normal. In this case, the group P is referred to as a Zappa–Szép product of G and H.
Free products
{{#invoke:mainmain}} The free product of G and H, usually denoted G ∗ H, is similar to the direct product, except that the subgroups G and H of G ∗ H are not required to commute. That is, if
 G = 〈 S_{G}  R_{G} 〉 and H = 〈 S_{H}  R_{H} 〉,
are presentations for G and H, then
 G ∗ H = 〈 S_{G} ∪ S_{H}  R_{G} ∪ R_{H} 〉.
Unlike the direct product, elements of the free product cannot be represented by ordered pairs. In fact, the free product of any two nontrivial groups is infinite. The free product is actually the coproduct in the category of groups.
Subdirect products
{{#invoke:mainmain}} If G and H are groups, a subdirect product of G and H is any subgroup of G × H which maps surjectively onto G and H under the projection homomorphisms. By Goursat's lemma, every subdirect product is a fiber product, and vice versa.
Fiber products
{{#invoke:mainmain}} Let G, H, and Q be groups, and let φ: G → Q and χ: H → Q be epimorphisms. The fiber product of G and H over Q, also known as a pullback, is the following subgroup of G × H:
 G ×_{Q} H = { (g, h) ∈ G × H : φ(g) = χ(h) }.
By Goursat's lemma, every subdirect product is a fiber product, and vice versa.
References
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