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 , 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.
Properties and related
- 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.