Group homomorphism
A group homomorphism is a morphism in Category of groups,
that is to say it is a structure-preserving map between groups.
#m/def/group Let
It immediately follows that
Proof
For the identity property, it is clear that
for any π ( π β π ) = π ( π ) = π ( π ) β’ π ( π ) , hence π β πΊ . For the latter property, notice that for any π ( π ) = π it follows π β πΊ , so π ( π β π β 1 ) = π ( π ) β’ π ( π β 1 ) = π ( π ) = π . π ( π β 1 ) = π ( π ) β 1
A bijective homomorphism is the a group isomorphism. Isomorphic groups have the same group table, and are essentially the same up to relabelling.
Properties and related
- Group monomorphism, Group epimorphism
- The Kernel of a group homomorphism
k e r β‘ ( π ) - The image
π ( πΊ ) π - A group homomorphism induces a subgroup homomorphism when its domain is restricted.