External direct product

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This article describes a product notion for groups

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Definition (for two groups)

Definition with symbols

Given two groups $G 1$ and $G 2$ , the external direct product of $G 1$ and $G 2$ , denoted as $G 1$ × $G 2$ , 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:

$(g 1, g 2) * (h 1, h 2) = (g 1 * h 1, g 2 * h 2)$

$(g_1, g_2)^{-1} = (g_1^{-1}, g_2^{-1})$

$e = (e, e)$

Equivalence with the internal direct product

Further information: Equivalence of internal and external direct product If $G = G_1 \times G_2$ is an external direct product, then the subgroups of $G$ given by $N 1 = G 1$ × $e$ and $N 2 = e$ × $G 2$ are normal subgroups of $G$ and $G$ is an internal direct product of these subgroups. 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 $G$ .

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 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 (including identity element, inverse and product) are performed coordinate-wise. Thus, the identity element is the element whose $i t h$ coordinate is the identity element of $G i$ . The inverse of an element is obtained by taking the inverse of each coordinate, and the product of two elements is obtained by coordinate-wise multiplication.

Note that for an infinite family, the external direct product does not correspond to an internal direct product, and the correct notion corresponding to internal direct product in that case is external direct sum (also called the restricted direct product).

Relation with other properties

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.

External direct product

From Groupprops

Contents

Definition (for two groups)

Definition with symbols

Equivalence with the internal direct product

Definition (for an infinite family of groups)

Relation with other properties

Weaker product notions

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