External direct product

Definition with symbols
Given two groups $$G_1$$ and $$G_2$$, the external direct product of $$G_1$$ and $$G_2$$, denoted as $$G_1 \times G_2$$, is defined as follows:


 * As a set, it is the Cartesian product of $$G_1$$ and $$G_2$$, that is, it is the set of ordered pairs $$(g_1, g_2)$$ with the first member $$g_1$$ from $$G_1$$ and the second member $$g_2$$ from $$G_2$$.
 * The group operations are defined coordinate-wise, that is:

Equivalence with the internal direct product
If $$G = G_1 \times G_2$$ is an external direct product, then the subgroups of $$G$$ given by $$N_1 = G_1 \times \{ e _2\}$$ and $$N_2 = \{ e_1 \} \times G_2$$ are normal subgroups of $$G$$ and $$G$$ is an internal direct product of these subgroups. (Here, $$e_1$$ is the identity element of $$G_1$$ and $$e_2$$ is the identity element of $$G_2$$). Conversely, any internal direct product of subgroups is isomorphic to their external direct product.

The two subgroups $$N_1$$ and $$N_2$$ are thus direct factors of $$G$$.

Natural projection maps to both direct factors
There are natural "projection" homomorphisms from the direct product $$G_1 \times G_2$$ to both the direct factors $$G_1$$ and $$G_2$$. Explicitly:


 * The projection $$\pi_1: G_1 \times G_2 \to G_1$$ is defined as $$(g_1,g_2) \mapsto g_1$$. The kernel of this homomorphism is the subgroup $$N_2 = \{ e_1 \} \times G_2$$.
 * The projection $$\pi_2: G_1 \times G_2 \to G_2$$ is defined as $$(g_1,g_2) \mapsto g_2$$. The kernel of this homomorphism is the subgroup $$N_1 = G_1 \times \{ e_2 \}$$.

Definition (for $$n \ge 2$$ groups)
Suppose $$G_1, G_2, \dots, G_n$$ are groups. The external direct product, denoted $$G_1 \times G_2 \times \dots \times G_n$$, is defined as follows:


 * As a set, it is the Cartesian product $$G_1 \times G_2 \times \dots \times G_n$$
 * The group operations are defined coordinate-wise:

Natural projection maps to all direct factors
For any $$i \in \{ 1,2,\dots,n \}$$, there is a natural "projection" homomorphism $$\pi_i: G_1 \times G_2 \times \dots \times G_n \to G_n$$ defined as:

$$(g_1,g_2,\dots,g_n) \mapsto g_i$$

The kernel of this homomorphism is the subgroup $$G_1 \times G_2 \times \dots \times G_{i-1} \times \{ e_i \} \times G_{i+1} \times \dots \times G_n$$, which is isomorphic to the external direct product of all the groups other than $$G_i$$.

Definition (for an infinite family of groups)
Suppose $$I$$ is an indexing set and $$\left\{ G_i \right \}_{i \in I}$$ is a family of groups. The external direct product of the $$G_i$$s, is defined as follows:


 * As a set, it is the Cartesian product of the $$G_i$$s
 * The group operations are as follows:

Natural projection maps to all direct factors
There is a natural projection map from the direct product to each direct factor. Explicitly, the projection $$\pi_j$$ to the direct factor $$G_j$$ is defined as:

$$\pi_j((g_i)_{i \in I}) = g_j$$

Definition as product in the category of groups
The external direct product of a family of groups, along with its natural coordinate projection maps to each of the groups, is the definition of product in the category of groups.

For two groups
Suppose $$G_1$$ and $$G_2$$ are groups. The categorical product of $$G_1$$ and $$G_2$$ would be defined as a group $$C$$ along with homomorphisms $$\pi_1:C \to G_1$$ and $$\pi_2:C \to G_2$$ such that for any group $$D$$ with homomorphisms $$f_1:D \to G_1, f_2:D \to G_2$$, there exists a unique homomorphism $$\varphi:D \to C$$ such that $$\pi_1 \circ \varphi = f_1$$ and $$\pi_2 \circ \varphi = f_2$$.

It is easy to see that the external direct product $$G_1 \times G_2$$ can be taken as $$C$$ with $$\pi_1$$ and $$\pi_2$$ being the natural projection maps $$(g_1,g_2) \mapsto g_1$$ and $$(g_1,g_2) \mapsto g_2$$ respectively.

Given a group $$D$$ with homomorphisms $$f_1:D \to G_1$$ and $$f_2:D \to G_2$$, the unique homomorphism $$\varphi$$ can be worked out to be:

$$\varphi(x) = (f_1(x),f_2(x)) \ \forall \ x \in D$$

Cancellation and factorization
A group (typically, a nontrivial group) is termed a directly indecomposable group if it is not isomorphic to the external direct product of two nontrivial groups. We have the following results related to direct factorization and indecomposable groups:

Single-valued arithmetic functions
Below we provide the information for a direct product of two groups. Information for a direct product of more than two groups can be inferred from this (for more, read the linked proof).

Effect on other constructs
We here identify $$G_1$$ with the subgroup $$G_1 \times \{ e_2 \}$$ inside $$G_1 \times G_2$$ by $$g \mapsto (g,e_2)$$ (where $$e_2$$ is the identity element. We also identify $$G_2$$ with the subgroup $$\{ e_1 \} \times G_2$$ inside $$G_1 \times G_2$$ by $$g \mapsto (e_1,g)$$.

Weaker product notions

 * Semidirect product which is set-theoretically a Cartesian product but for which the group-theoretical multiplication has a twist on one of the factors
 * Exact factorization which a set-theoretically a Cartesian product but for which the group-theoretical multiplication has a twist on both of the factors
 * Group extension which could be viewed as a set-theoretic direct product with correction in terms of cocycles.