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External direct product
From Groupprops
This article describes a product notion 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 G1 and G2, the external direct product of G1 and G2, denoted as G1 × G2, is defined as follows:
- As a set, it is the Cartesian product of G1 and G2, that is, it is the set of ordered pairs (g1,g2) with the first member g1 from G1 and the second member (g2) from G2.
- The group operations are defined coordinate-wise, that is:
-
.
-
.
-
.
-
Here, e1 is the identity element for G1 and e2 is the identity element for G2.
Equivalence with the internal direct product
Further information: Equivalence of internal and external direct product
If
is an external direct product, then the subgroups of G given by
and
are normal subgroups of G and G is an internal direct product of these subgroups. (Here, e1 is the identity element of G1 and e2 is the identity element of G2). Conversely, any internal direct product of subgroups is isomorphic to their external direct product.
The two subgroups N1 and N2 are thus direct factors of G.
Definition (for
groups)
Suppose
are groups. The external direct product, denoted
, is defined as follows:
- As a set, it is the Cartesian product
- The group operations are defined coordinate-wise:
Here, ei is the identity element in Gi.
Definition (for an infinite family of groups)
Suppose I is an indexing set and
is a family of groups. The direct product of the Gis is defined as follows:
- As a set, it is the Cartesian product of the Gis
- The group operations (including identity element, inverse and product) are performed coordinate-wise. Thus, the identity element is the element whose ith coordinate is the identity element of Gi. 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 | G1 has order a1, G2 has order a2 | has order a1a2 | order of direct product is product of orders |
| exponent | G1 has exponent a1, G2 has order a2 | has order | exponent of direct product is lcm of exponents |
| minimum size of generating set | G1 has minimum size of generating set equal to a1, G2 has minimum size of generating set equal to a2 | has minimum size of generating set at most a1 + a2, and at least max(a1,a2) | 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 | G1 nilpotent of class c1, G2 nilpotent of class c2 | is nilpotent of class max(c1,c2) | nilpotency of fixed class is direct product-closed |
| derived length | G1 solvable of derived length l1, G2 solvable of derived length l2 | solvable of derived length max(l1,l2) | (similar to proof for nilpotency) |
| Fitting length | G1 has Fitting length a1, G2 has Fitting length a2 | has Fitting length max(a1,a2) | |
| Frattini length | G1 has Frattini length a1, G2 has Frattini length a2 | has Frattini length max(a1,a2) | |
| Composition length | G1 has composition length a1, G2 has composition length a2 | has composition length a1 + a2 | |
| Chief length | G1 has chief length a1, G2 has chief length a2 | has chief length a1 + a2 | |
| Number of conjugacy classes | G1 has a1 conjugacy classes, G2 has a2 conjugacy classes | has a1a2 conjugacy classes | |
| Number of subgroups | G1 has a1 subgroups, G2 has a2 subgroups | has at least a1a2 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 G1 and a conjugacy class size in G2. If there are ai conjugacy classes in Gi, we get a1a2 products | |
| degrees of irreducible representations | We take every possible product of a degree of irreducible representation of G1 and a degree of irreducible representation of G2. If there are ai irreducible representations of Gi, we get a1a2 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 (d1,d2) with lcm d of the product of the number of elements of order d1 in G1 and the number of elements of order d2 in G2 |
Effect on other constructs
We here identify G1 with the subgroup
inside
by
(where e2 is the identity element. We also identify G2 with the subgroup
inside
by
.
| Construct | Behavior/value on direct product in terms of behavior/value on G1 and G2 | Proof |
|---|---|---|
| generating set | we can take the union of the generating set values for G1 and for G2 | |
| presentation of a group | we take the union of the generating sets for G1 and G2, the union of the relators for G1 and G2, and additional relations stating that each generator for G1 commutes with each generator for G2 | |
| irreducible representations | For each irreducible representation of G1 and each irreducible representation of G2, we take the tensor product to get an irreducible representation of | 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.