Group homomorphism

Written by Patrick Stevens, et al. last updated
Requires: Function, Group

A group homomorphism is a function between groups which "respects the group structure".

Definition

Formally, given two groups and (which hereafter we will abbreviate as and respectively), a group homomorphism from to is a function from the underlying set to the underlying set , such that for all .

Examples

  • For any group , there is a group homomorphism , given by for all . This homomorphism is always bijective.
  • For any group , there is a (unique) group homomorphism into the group with one element and the only possible group operation . This homomorphism is given by for all . This homomorphism is usually not injective: it is injective if and only if is the group with one element. (Uniqueness is guaranteed because there is only one function, let alone group homomorphism, from any set to a set with one element.)
  • For any group , there is a (unique) group homomorphism from the group with one element into , given by , the identity of . This homomorphism is usually not surjective: it is surjective if and only if is the group with one element. (Uniqueness is guaranteed this time by the property proved below that the identity gets mapped to the identity.)
  • For any group , there is a bijective group homomorphism to another group given by taking inverses: . The group is defined to have underlying set equal to that of , and group operation .
  • For any pair of groups , there is a homomorphism between and given by .
  • There is only one homomorphism between the group with two elements and the group with three elements; it is given by . For example, the function given by is not a group homomorphism, because if it were, then , which is not true. (We have used that the identity gets mapped to the identity.)

Properties

  • The identity gets mapped to the identity. (Proof.)
  • The inverse of the image is the image of the inverse. (Proof.)
  • The image of a group under a homomorphism is another group. (Proof.)
  • The composition of two homomorphisms is a homomorphism. (Proof.)