Homomorphic image of subgroup is subgroup

Statement

Let $G$, $G'$ be groups and $\phi :G\to G'$ be a homomorphism between them. Suppose $H\leq G$ is a subgroup of $G$. Then $H'=\phi (H)$ is a subgroup of $G'$.

Proof

We check that $H'$ satisfies the group axioms.

Identity element

$e$ is the identity of $H$ implies $\phi (e)$ is the identity of $H'$.

Closure

Every pair of elements of $H'$ can be written as $\phi (h_{1})$, $\phi (h_{2})$ for $h_{1},h_{2}\in H$. Then $\phi (h_{1})\phi (h_{2})=\phi (h_{1}h_{2})\in H'$ since $\phi$ is a homomorphism.

Inverses

The inverse of $\phi (h)\in H'$ is $\phi (h)^{-1}=\phi (h^{-1})$.

Associativity

This is inherited from $G'$.