A group homomorphism induces a subgroup homomorphism

Given a Group homomorphism $𝜙 : 𝐺 \to 𝐺'$ and a subgroup $𝐻 \subseteq 𝐺$ , the map $𝜙 𝐻 : 𝐻 \to 𝐺' : ℎ \mapsto 𝜙 (ℎ)$ is also a homomorphism. #m/thm/group Quite trivial.

Proof

$𝜙$ is a group homomorphism iff $𝜙 (𝑎 𝑏) = 𝜙 (𝑎) 𝜙 (𝑏)$ for all $𝑎, 𝑏 \in 𝐺$ . Clearly if $𝑎, 𝑏 \in 𝐻 \subseteq 𝐺$ , the property still holds, hence $𝜙'$ is a homomorphism.

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.

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