Direct product of groups

Definition

Given groups G and H, the direct product G × H is defined as follows:

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.

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.)

1. The intersection G ∩ H is trivial.
2. Every element of P can be expressed as the product of an element of G and an element of H.
3. 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.

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 | a³ = 1 〉     and     H = 〈 b | b⁵ = 1 〉.

Then

G × H = 〈 a, b | a³ = 1, b⁵ = 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.

Universal property

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 of the direct products is isomorphic to the direct product of the quotients:

(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.

The subgroups of direct products are described by Goursat's lemma. Other subgroups include fiber products of G and H.

Conjugacy and centralizers

Two elements (g₁, h₁) and (g₂, h₂) are conjugate in G × H if and only if g₁ and g₂ are conjugate in G and h₁ and h₂ 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(g₁, g₂) = (g₂, g₁).

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

[ α β γ δ ]

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.

Finite direct products

It is possible to take the direct product of more than two groups at once. Given a finite sequence G₁, ..., G_n of groups, the direct product

∏ i = 1 n G i = G 1 × G 2 × ⋯ × G n

is defined as follows:

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 G₁, G₂, … 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:

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 non-identity components.

Semidirect products

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:

1. The intersection G ∩ H is trivial.
2. Every element of P can be expressed as the product of an element of G and an element of H.
3. 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

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

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.

Fiber products

Let G, H, and Q be groups, and let φ: G → Q and χ: H → Q be homomorphisms. 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) }.

If φ: G → Q and χ: H → Q are epimorphism, then this is a subdirect product.