Algebraic structure → Group theory Group theory |
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In mathematics, specifically 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 group-theoretic 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 . 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.
Given groups G (with operation *) and H (with operation ∆), the direct product G × H is defined as follows:
The resulting algebraic object satisfies the axioms for a group. Specifically:
* | 1 | a |
---|---|---|
1 | 1 | a |
a | a | 1 |
* | 1 | b |
---|---|---|
1 | 1 | b |
b | b | 1 |
Then the direct product G × H is isomorphic to the Klein four-group:
* | (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) |
Let G and H be groups, let P = G × H, and consider the following two subsets of P:
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.)
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:
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.
∙ | 1 | a | b | c |
---|---|---|---|---|
1 | 1 | a | b | c |
a | a | 1 | c | b |
b | b | c | 1 | a |
c | c | b | a | 1 |
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
where and are (disjoint) generating sets and and are defining relations. Then
where is a set of relations specifying that each element of commutes with each element of .
For example if
then
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
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.
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
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, 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:
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 non-trivial group, then the product G × G has a diagonal subgroup
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.
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:
Similarly, the center of G × H is the product of the centers of G and H:
Normalizers behave in a more complex manner since not all subgroups of direct products themselves decompose as direct products.
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
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.
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.
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:
This has many of the same properties as the direct product of two groups, and can be characterized algebraically in a similar way.
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 Π_{i∈I} G_{i} is defined as follows:
Unlike a finite direct product, the infinite direct product Π_{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.
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:
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.
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
are presentations for G and H, then
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.
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.
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:
If φ: G → Q and χ: H → Q are epimorphisms, then this is a subdirect product.