External direct product

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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_1h_1, g_2h_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 \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:
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_1h_1,g_2h_2,\dots,g_nh_n) where g_i,h_i \in G_i, 1 \le i \le 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 \le i \le 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_is, is defined as follows:

  • As a set, it is the Cartesian product of the G_is
  • 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_ih_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_1a_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_1a_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_1a_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_1a_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_1a_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.