Group homomorphism

A group homomorphism is a morphism in , that is to say it is a structure-preserving map between groups. #m/def/group Let and be groups, and let . Then is a homomorphism iff for any

It immediately follows that and .

Proof

For the identity property, it is clear that for any , hence . For the latter property, notice that for any it follows , so .

A bijective homomorphism is the a group isomorphism. Isomorphic groups have the same group table, and are essentially the same up to relabelling.

Group monomorphism, Group epimorphism
The Kernel of a group homomorphism is the set of all domain elements that map to the identity, and it forms a normal subgroup (proof in Zettel)
The image is the range of , and The image of a group homomorphism is a subgroup.
A group homomorphism induces a subgroup homomorphism when its domain is restricted.

#state/tidy | #lang/en | #SemBr

Group homomorphism

Properties and related