External direct product

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This article describes a product notion for groups

1 Definition (for two groups)
- 1.1 Definition with symbols
- 1.2 Equivalence with the internal direct product
2 Definition (for groups)
3 Definition (for an infinite family of groups)
4 Effect on arithmetic functions
- 4.1 Single-valued arithmetic functions
- 4.2 Lists/multisets
5 Effect on other constructs
6 Relation with other product notions
- 6.1 Weaker product notions

This article is about a basic definition in group theory.The article text may, however, contain advanced material.
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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:
- $\! (g_1, g_2) * (h_1, h_2) = (g_1 * h_1, g_2 * h_2)$ .
- $(g_1, g_2)^{-1} = (g_1^{-1}, g_2^{-1})$ .
- $\! e = (e_1, e_2)$ .

Here, $e 1$ is the identity element for $G 1$ and $e 2$ is the identity element for $G 2$ .

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 $N_1 = G_1 \times \{ e _2\}$ 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 $G$ .

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:
- $\! (g_1, g_2, \dots, g_n)(h_1,h_2,\dots, h_n) = (g_1h_1,g_2h_2,\dots,g_nh_n)$
- $\! (g_1,g_2,\dots,g_n)^{-1} = (g_1^{-1},g_2^{-1},\dots,g_n^{-1})$
- $\! e = (e_1,e_2,\dots,e_n)$

Here, $e i$ is the identity element in $G i$ .

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 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).

Effect on arithmetic functions

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.

Arithmetic function	Values at input groups	Value on direct product	Proof
order	$G 1$ has order $a 1$ , $G 2$ has order $a 2$	$G_1 \times G_2$ has order $a 1 a 2$	order of direct product is product of orders
exponent	$G 1$ has exponent $a 1$ , $G 2$ has order $a 2$	$G_1 \times G_2$ has order $\operatorname{lcm}(a_1,a_2)$	exponent of direct product is lcm of exponents
minimum size of generating set	$G 1$ has minimum size of generating set equal to $a 1$ , $G 2$ has minimum size of generating set equal to $a 2$	$G_1 \times G_2$ has minimum size of generating set at most $a 1 + a 2$ , and at least $max(a 1, a 2)$	Minimum size of generating set of direct product of two groups is bounded by sum of minimum size of generating set for each (follow through for counterexamples too)
nilpotency class	$G 1$ nilpotent of class $c 1$ , $G 2$ nilpotent of class $c 2$	$G_1 \times G_2$ is nilpotent of class $max(c 1, c 2)$	nilpotency of fixed class is direct product-closed
derived length	$G 1$ solvable of derived length $l 1$ , $G 2$ solvable of derived length $l 2$	$G_1 \times G_2$ solvable of derived length $max(l 1, l 2)$	(similar to proof for nilpotency)
Fitting length	$G 1$ has Fitting length $a 1$ , $G 2$ has Fitting length $a 2$	$G_1 \times G_2$ has Fitting length $max(a 1, a 2)$
Frattini length	$G 1$ has Frattini length $a 1$ , $G 2$ has Frattini length $a 2$	$G_1 \times G_2$ has Frattini length $max(a 1, a 2)$
Composition length	$G 1$ has composition length $a 1$ , $G 2$ has composition length $a 2$	$G_1 \times G_2$ has composition length $a 1 + a 2$
Chief length	$G 1$ has chief length $a 1$ , $G 2$ has chief length $a 2$	$G_1 \times G_2$ has chief length $a 1 + a 2$
Number of conjugacy classes	$G 1$ has $a 1$ conjugacy classes, $G 2$ has $a 2$ conjugacy classes	$G_1 \times G_2$ has $a 1 a 2$ conjugacy classes
Number of subgroups	$G 1$ has $a 1$ subgroups, $G 2$ has $a 2$ subgroups	$G_1 \times G_2$ has at least $a 1 a 2$ subgroups

Lists/multisets

Arithmetic function	How we obtain value on direct product	Proof
sizes of conjugacy classes (as a multiset)	We take every possible product of a conjugacy class size in $G 1$ and a conjugacy class size in $G 2$ . If there are $a i$ conjugacy classes in $G i$ , we get $a 1 a 2$ products
degrees of irreducible representations	We take every possible product of a degree of irreducible representation of $G 1$ and a degree of irreducible representation of $G 2$ . If there are $a i$ irreducible representations of $G i$ , we get $a 1 a 2$ products	Tensor product establishes bijection between irreducible representations of direct factors and direct product
order statistics	The number of elements of order $d$ in the direct product is the sum over all pairs $(d 1, d 2)$ with lcm $d$ of the product of the number of elements of order $d 1$ in $G 1$ and the number of elements of order $d 2$ in $G 2$

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 $g \mapsto (e_1,g)$ .

Construct	Behavior/value on direct product $G_1 \times G_2$ in terms of behavior/value on $G 1$ and $G 2$	Proof
generating set	we can take the union of the generating set values for $G 1$ and for $G 2$
presentation of a group	we take the union of the generating sets for $G 1$ and $G 2$ , the union of the relators for $G 1$ and $G 2$ , and additional relations stating that each generator for $G 1$ commutes with each generator for $G 2$
irreducible representations	For each irreducible representation of $G 1$ and each irreducible representation of $G 2$ , we take the tensor product to get an irreducible representation of $G_1 \times G_2$	Tensor product establishes bijection between irreducible representations of direct factors and direct product

Relation with other product notions

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.

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External direct product

From Groupprops

Contents

Definition (for two groups)

Definition with symbols

Equivalence with the internal direct product

Definition (for $n \ge 2$ groups)

Definition (for an infinite family of groups)

Effect on arithmetic functions

Single-valued arithmetic functions

Lists/multisets

Effect on other constructs

Relation with other product notions

Weaker product notions

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Variants

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External direct product

From Groupprops

Contents

Definition (for two groups)

Definition with symbols

Equivalence with the internal direct product

Definition (for groups)

Definition (for an infinite family of groups)

Effect on arithmetic functions

Single-valued arithmetic functions

Lists/multisets

Effect on other constructs

Relation with other product notions

Weaker product notions

Personal tools

Namespaces

Variants

Views

Actions

Search

Navigation

lookup

Credits

Toolbox

request/feedback

subject wikis

Definition (for $n \ge 2$ groups)