A group homomorphism induces a subgroup homomorphism
Given a Group homomorphism
Proof
is a group homomorphism iff π for all π ( π π ) = π ( π ) π ( π ) . Clearly if π , π β πΊ , the property still holds, hence π , π β π» β πΊ is a homomorphism. π β²
Corollary
It follows that a group automorphism establishes isomorphisms between subgroups, since the the image of a group homomorphism is a subgroup, which for an automorphism will be a subgroup of equal size.