External direct product

This article describes a product notion for groups. See other related product notions for groups.

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 $G_{1}\times G_{2}$, 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:

Operation name	Description of operation in terms of description of operations on factor groups	Explanation
Multiplication or product	$(g_{1},g_{2})(h_{1},h_{2})=(g_{1}h_{1},g_{2}h_{2})$ where $g_{1},h_{1}\in G_{1}$, $g_{2},h_{2}\in G_{2}$	We carry out the multiplication separately in each coordinate.
Identity element (or neutral element)	$\!e=(e_{1},e_{2})$ where $e_{1}$ is the identity element of $G_{1}$ and $e_{2}$ is the identity element of $G_{2}$.	To compute the identity element, we use the identity element in each coordinate.
Inverse map	$(g_{1},g_{2})^{-1}=(g_{1}^{-1},g_{2}^{-1})$	We carry out the inversion separately in each coordinate.

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 $G$.

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 $n\geq 2$ 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:

Operation name	Description of operation in terms of description of operations on factor groups	Explanation
Multiplication or product	$(g_{1},g_{2},\dots ,g_{n})(h_{1},h_{2},\dots ,h_{n})=(g_{1}h_{1},g_{2}h_{2},\dots ,g_{n}h_{n})$ where $g_{i},h_{i}\in G_{i},1\leq i\leq n$.	We multiply separately in each coordinate.
Identity element	$\!e=(e_{1},e_{2},\dots ,e_{n})$ where $e_{i}$ is the identity element of $G_{i}$ for $1\leq i\leq n$.	To compute the identity element, we use the identity element in each coordinate.
Inverse element	$\!(g_{1},g_{2},\dots ,g_{n})^{-1}=(g_{1}^{-1},g_{2}^{-1},\dots ,g_{n}^{-1})$	To compute the inverse, we calculate the inverse in each coordinate.

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:

Operation name	Description of operation in terms of description of operations on factor groups
Multiplication or product	The product of $g=(g_{i})_{i\in I}$ and $h=(h_{i})_{i\in I}$ is $gh=(g_{i}h_{i})_{i\in I}$
Identity element	The identity element $e$ is the element $(e_{i})_{i\in I}$ where $e_{i}$ is the identity element of $G_{i}$.
Inverse element	The inverse of $g=(g_{i})_{i\in I}$ is the element $g^{-1}=(g_{i}^{-1})_{i\in I}$.

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:

Name	Statement
direct product is cancellative for finite groups	If $G,H,K$ are finite groups, and $G\times H\cong G\times K$, then $H\cong K$.
Krull-Remak-Schmidt theorem	Any group of finite composition length (equivalently, a group satisfying ascending chain condition on normal subgroups and group satisfying descending chain condition on normal subgroups) has an essentially unique factorization as a direct product of directly indecomposable groups.
Corollary of Krull-Remak-Schmidt theorem for cancellation of factors in direct product	If $G,H,K$ are groups of finite composition length, and $G\times H\cong G\times K$, then $H\cong K$.
Corollary of Krull-Remak-Schmidt theorem for cancellation of powers	If $G$ and $H$ are groups of finite composition length and $m$ is a positive integer such that $G^{m}\cong H^{m}$, then $G\cong H$.

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 (for more, read the linked proof).

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 is bounded below by maximum of minimum size of generating set of each factor minimum size of generating set of direct product is bounded by sum of minimum size of generating set of each factor
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 class of direct product is maximum of nilpotency classes
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 $\max(l_{1},l_{2})$	derived length of direct product is maximum of derived lengths
Fitting length	$G_{1}$ has Fitting length $a_{1}$, $G_{2}$ has Fitting length $a_{2}$	$G_{1}\times G_{2}$ has Fitting length $\max(a_{1},a_{2})$	Fitting length of direct product is maximum of Fitting lengths
Frattini length	$G_{1}$ has Frattini length $a_{1}$, $G_{2}$ has Frattini length $a_{2}$	$G_{1}\times G_{2}$ has Frattini length $\max(a_{1},a_{2})$	Frattini length of direct product is maximum of Frattini lengths
Composition length	$G_{1}$ has composition length $a_{1}$, $G_{2}$ has composition length $a_{2}$	$G_{1}\times G_{2}$ has composition length $a_{1}+a_{2}$	composition length of direct product is sum of composition lengths
Chief length	$G_{1}$ has chief length $a_{1}$, $G_{2}$ has chief length $a_{2}$	$G_{1}\times G_{2}$ has chief length $a_{1}+a_{2}$	chief length of direct product is sum of chief lengths
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 conjugacy classes in a direct product is the product of the number of conjugacy classes in each factor
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	number of subgroups of direct product is bounded below by product of number of subgroups in each factor

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	conjugacy class sizes of direct product are pairwise products of conjugacy class sizes of direct factors
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	degrees of irreducible representations of direct product are pairwise products of degrees of irreducible representations of direct factors This follows from 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 $G_{2}$. If we use cumulative order statistics instead, the number of elements of order dividing $d$ in $G_{1}\times G_{2}$ is the product of the number of elements of order dividing $d$ in $G_{1}$ and the number of elements of order dividing $d$ in $G_{2}$.	cumulative order statistics of direct product is obtained by taking pointwise products of cumulative order statistics of direct factors

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 $G_{2}$	presentation of direct product is disjoint union of presentations plus commutation relations
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
encoding of a group	We can combine encodings of $G_{1}$ and $G_{2}$ to obtain an encoding of $G_{1}\times G_{2}$.	Encoding of external direct product in terms of encodings of direct factors

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.