Direct product of groups

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 $$G \oplus H$$. 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 $G$(with operation $$) and $H$ (with operation $∆$), the direct product $G × H$ is defined as follows:1. The underlying set is the Cartesian product, $G × H$. That is, the ordered pairs $(g, h)$, where $g ∈ G$ and $h ∈ H$.

2. The binary operation on $G × H$ is defined component-wise:

The resulting algebraic object satisfies the axioms for a group. Specifically:
 * Associativity: The binary operation on $(g_{1}, h_{1}) · (g_{2}, h_{2}) = (g_{1} * g_{2}, h_{1} ∆ h_{2})$ is associative.
 * Identity: The direct product has an identity element, namely $G × H$, where $(1_{G}, 1_{H})$ is the identity element of $1_{G}$ and $G$ is the identity element of $1_{H}$.
 * Inverses: The inverse of an element $H$ of $(g, h)$ is the pair $G × H$, where $(g^{−1}, h^{−1})$ is the inverse of $g^{−1}$ in $g$, and $G$ is the inverse of $h^{−1}$ in $h$.

Examples

 * Let $H$ be the group of real numbers under addition. Then the direct product $R$ is the group of all two-component vectors $R × R$ under the operation of vector addition:


 * Let $(x, y)$ be the group of positive real numbers under multiplication. Then the direct product $(x_{1}, y_{1}) + (x_{2}, y_{2}) = (x_{1} + x_{2}, y_{1} + y_{2})$ is the group of all vectors in the first quadrant under the operation of component-wise multiplication

      Then the direct product $R^{+}$ is isomorphic to the Klein four-group:
 * Let $R^{+} × R^{+}$ and $(x_{1}, y_{1}) × (x_{2}, y_{2}) = (x_{1} × x_{2},  y_{1} × y_{2})$ be cyclic groups with two elements each:

Algebraic structure
Let $G$ and $H$ be groups, let $G × H$, and consider the following two subsets of $G × H ≅ H × G$:


 * $(G × H) × K ≅ G × (H × K)$   and    $G$.

Both of these are in fact subgroups of $H$, the first being isomorphic to $K$, and the second being isomorphic to $E$. If we identify these with $G ≅ G × E ≅ E × G$ and $G$, respectively, then we can think of the direct product $G × H$ as containing the original groups $G$ and $H$ as subgroups.

These subgroups of $|G × H| = |G| |H|$ have the following three important properties: (Saying again that we identify $(g, h)$ and $g$ with $h$ and $|(g, h)| = lcm(|g|, |h|)$, respectively.)

Together, these three properties completely determine the algebraic structure of the direct product $| g |$. That is, if $| h |$ is any group having subgroups $(g, h)$ and $g$ that satisfy the properties above, then $h$ is necessarily isomorphic to the direct product of $G$ and $H$. In this situation, $G × H$ is sometimes referred to as the internal direct product of its subgroups $m$ and $n$.
 * 1) The intersection $(Z / mZ) × (Z / nZ) ≅ Z / mnZ$ is trivial.
 * 2) Every element of $G$ can be expressed uniquely as the product of an element of $H$ and an element of $P = G × H$.
 * 3) Every element of $P$ commutes with every element of $G′ = { (g, 1) : g ∈ G }$.

In some contexts, the third property above is replaced by the following:
 * 3′. Both $H′ = { (1, h) : h ∈ H }$ and $P$ are normal in $G$.

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 $H$ of any $G$ in $H$, $P$ in $G$.

Presentations
The algebraic structure of $H$ can be used to give a presentation for the direct product in terms of the presentations of $P$ and $G′$. Specifically, suppose that


 * $$G = \langle S_G \mid R_G \rangle \ \ $$ and $$\ \ H = \langle S_H \mid R_H \rangle,$$

where $$S_G$$ and $$S_H$$ are (disjoint) generating sets and $$R_G$$ and $$R_H$$ are defining relations. Then


 * $$G \times H = \langle S_G \cup S_H \mid R_G \cup R_H \cup R_P \rangle$$

where $$R_P$$ is a set of relations specifying that each element of $$S_G$$ commutes with each element of $$S_H$$.

For example if


 * $$G = \langle a \mid a^3=1 \rangle \ \ $$ and $$\ \ H = \langle b \mid b^5=1 \rangle$$

then


 * $$G \times H = \langle a, b \mid a^3 = 1, b^5 = 1, ab=ba \rangle.$$

Normal structure
As mentioned above, the subgroups $H′$ and $G$ are normal in $H$. Specifically, define functions $G ∩ H$ and $P$ by


 * $G$    and     $H$.

Then $G$ and $H$ are homomorphisms, known as projection homomorphisms, whose kernels are $P$ and $P$, respectively.

It follows that $G$ is an extension of $H$ by $P$ (or vice versa). In the case where $G$ is a finite group, it follows that the composition factors of $H$ are precisely the union of the composition factors of $P$ and the composition factors of $G$.

Universal property
The direct product $H$ can be characterized by the following universal property. Let $G$ and $H$ be the projection homomorphisms. Then for any group $P$ and any homomorphisms $[g,h]$ and $g$, there exists a unique homomorphism $G$ making the following diagram commute:
 * [[Image:DirectProductDiagram.png|200px]]

Specifically, the homomorphism $h$ is given by the formula

This is a special case of the universal property for products in category theory.

Subgroups
If $H$ is a subgroup of $V$ and $V$ is a subgroup of $1$, then the direct product $a$ is a subgroup of $b$. For example, the isomorphic copy of $c$ in $1$ is the product $1$, where $a$ is the trivial subgroup of $b$.

If $c$ and $a$ are normal, then $a$ is a normal subgroup of $1$. 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 $c$ is the product of a subgroup of $b$ with a subgroup of $b$. For example, if $b$ is any non-trivial group, then the product $c$ has a diagonal subgroup

which is not the direct product of two subgroups of $1$.

The subgroups of direct products are described by Goursat's lemma. Other subgroups include fiber products of $a$ and $c$.

Conjugacy and centralizers
Two elements $c$ and $b$ are conjugate in $a$ if and only if $1$ and $V$ are conjugate in $C^{×}$ and $C^{×}$ and $T$ are conjugate in $R^{+}$. It follows that each conjugacy class in $GL(n, R)$ is simply the Cartesian product of a conjugacy class in $SL(n, R)$ and a conjugacy class in $O(n, R)$.

Along the same lines, if $SO(n, R)$, the centralizer of ${−I, I},$ is simply the product of the centralizers of $I$ and ${−I, I},$:



Similarly, the center of $I$ is the product of the centers of $−I$ and $G × H$:



Normalizers behave in a more complex manner since not all subgroups of direct products themselves decompose as direct products.

Automorphisms and endomorphisms
If $G$ is an automorphism of $H$ and $G$ is an automorphism of $H$, then the product function $G × H$ defined by



is an automorphism of $π_{G}: G × H → G$. It follows that $π_{H}: G × H → H$ has a subgroup isomorphic to the direct product $π_{G}(g, h) = g$.

It is not true in general that every automorphism of $π_{H}(g, h) = h$ has the above form. (That is, $π_{G}$ is often a proper subgroup of $π_{H}$.) For example, if $H$ is any group, then there exists an automorphism $G$ of $G × H$ that switches the two factors, i.e.



For another example, the automorphism group of $G$ is $H$, the group of all $G × H$ matrices with integer entries and determinant, $G × H$. This automorphism group is infinite, but only finitely many of the automorphisms have the form given above.

In general, every endomorphism of $G$ can be written as a $H$ matrix


 * $$\begin{bmatrix}\alpha & \beta \\ \gamma & \delta\end{bmatrix}$$

where $G × H$ is an endomorphism of $π_{G}: G × H → G$, $π_{H}: G × H → H$ is an endomorphism of $P$, and $ƒ_{G}: P → G$ and $ƒ_{H}: P → H$ are homomorphisms. Such a matrix must have the property that every element in the image of $ƒ: P → G × H$ commutes with every element in the image of $ƒ$, and every element in the image of $ƒ(p) = ( ƒ_{G}(p),  ƒ_{H}(p) )$ commutes with every element in the image of $A$.

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 $G$ of groups, the direct product


 * $$\prod_{i=1}^n G_i \;=\; G_1 \times G_2 \times \cdots \times G_n$$

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 $B$ 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 { $H$ }$A × B$ of groups, the direct product $G × H$ is defined as follows:

Unlike a finite direct product, the infinite direct product $G$ is not generated by the elements of the isomorphic subgroups { $G × H$ }$G × {1}$. 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 ${1}$ with subgroups $H$ and $A$ is isomorphic to the direct product of $B$ and $A × B$ as long as it satisfies the following three conditions: A semidirect product of $G × H$ and $(G × H) / (A × B) ≅ (G / A) × (H / B)$ 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 $G$, but with a slightly more complicated rule for multiplication.
 * 1) The intersection $H$ is trivial.
 * 2) Every element of $G$ can be expressed uniquely as the product of an element of $G × G$ and an element of $Δ = { (g, g) : g ∈ G }$.
 * 3) Both $G$ and $G$ are normal in $H$.

It is also possible to relax the third condition entirely, requiring neither of the two subgroups to be normal. In this case, the group $(g_{1}, h_{1})$ is referred to as a Zappa–Szép product of $(g_{2}, h_{2})$ and $G × H$.

Free products
The free product of $g_{1}$ and $g_{2}$, usually denoted $G$, is similar to the direct product, except that the subgroups $h_{1}$ and $h_{2}$ of $H$ are not required to commute. That is, if


 * $G × H$ = $G$ | $H$    and     $(g, h) ∈ G × H$ = $(g, h)$ | $g$,

are presentations for $h$ and $C_{G×H}(g, h)$, 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.

Subdirect products
If $C_{G}(g) × C_{H}(h)$ and $G × H$ are groups, a subdirect product of $G$ and $H$ is any subgroup of $Z(G × H)$ which maps surjectively onto $Z(G) × Z(H)$ and $α$ under the projection homomorphisms. By Goursat's lemma, every subdirect product is a fiber product.

Fiber products
Let $G$, $β$, and $H$ be groups, and let $α × β: G × H → G × H$ and $(α × β)(g, h) = ( α(g), β(h) )$ be homomorphisms. The fiber product of $G × H$ and $Aut(G × H)$ over $Aut(G) × Aut(H)$, also known as a pullback, is the following subgroup of $G × H$:

$$G \times_{Q} H = \{ \, (g, h) \in G \times H : \varphi(g) = \chi(h) \,\}\text{.}$$ If $Aut(G) × Aut(H)$ and $Aut(G × H)$ are epimorphisms, then this is a subdirect product.