# External direct product

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

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## Definition (for two groups)

### Definition with symbols

Given two groups and , the **external direct product** of and , denoted as , is defined as follows:

- As a set, it is the Cartesian product of and , that is, it is the set of
*ordered pairs*with the first member from and the second member from . - 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 | where , | We carry out the multiplication separately in each coordinate. |

Identity element (or neutral element) | where is the identity element of and is the identity element of . | To compute the identity element, we use the identity element in each coordinate. |

Inverse map | 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 is an external direct product, then the subgroups of given by and are normal subgroups of and is an internal direct product of these subgroups. (Here, is the identity element of and is the identity element of ). Conversely, any internal direct product of subgroups is isomorphic to their external direct product.

The two subgroups and are thus direct factors of .

### Natural projection maps to both direct factors

There are natural "projection" homomorphisms from the direct product to both the direct factors and . Explicitly:

- The projection is defined as . The kernel of this homomorphism is the subgroup .
- The projection is defined as . The kernel of this homomorphism is the subgroup .

## 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:

Operation name | Description of operation in terms of description of operations on factor groups | Explanation |
---|---|---|

Multiplication or product | where . | We multiply separately in each coordinate. |

Identity element | where is the identity element of for . | To compute the identity element, we use the identity element in each coordinate. |

Inverse element | To compute the inverse, we calculate the inverse in each coordinate. |

### Natural projection maps to all direct factors

For any , there is a natural "projection" homomorphism defined as:

The kernel of this homomorphism is the subgroup , which is isomorphic to the external direct product of all the groups other than .

## Definition (for an infinite family of groups)

Suppose is an indexing set and is a family of groups. The external direct product of the s, is defined as follows:

- As a set, it is the Cartesian product of the 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 and is |

Identity element | The identity element is the element where is the identity element of . |

Inverse element | The inverse of is the element . |

### Natural projection maps to all direct factors

There is a natural projection map from the direct product to each direct factor. Explicitly, the projection to the direct factor is defined as:

## 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 and are groups. The *categorical product* of and would be defined as a group along with homomorphisms and such that for any group with homomorphisms , there exists a unique homomorphism such that and .

It is easy to see that the external direct product can be taken as with and being the natural projection maps and respectively.

Given a group with homomorphisms and , the unique homomorphism can be worked out to be:

## 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 are finite groups, and , then . |

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 are groups of finite composition length, and , then . |

Corollary of Krull-Remak-Schmidt theorem for cancellation of powers | If and are groups of finite composition length and is a positive integer such that , then . |

## 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).

### 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 and a conjugacy class size in . If there are conjugacy classes in , we get 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 and a degree of irreducible representation of . If there are irreducible representations of , we get 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 in the direct product is the sum over all pairs with lcm of the product of the number of elements of order in and the number of elements of order in . If we use cumulative order statistics instead, the number of elements of order dividing in is the product of the number of elements of order dividing in and the number of elements of order dividing in . |
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 with the subgroup inside by (where is the identity element. We also identify with the subgroup inside by .

Construct | Behavior/value on direct product in terms of behavior/value on and | Proof |
---|---|---|

generating set | we can take the union of the generating set values for and for | |

presentation of a group | we take the union of the generating sets for and , the union of the relators for and , and additional relations stating that each generator for commutes with each generator for | presentation of direct product is disjoint union of presentations plus commutation relations |

irreducible representations | For each irreducible representation of and each irreducible representation of , we take the tensor product to get an irreducible representation of | Tensor product establishes bijection between irreducible representations of direct factors and direct product |

encoding of a group | We can combine encodings of and to obtain an encoding of . | 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.