I have a question related to the following property of group homomorphisms.
Let $f:G \rightarrow G'$ be a homomorphism between two groups.
If $H'$ is a subgroup of $G'$, then $f^{-1}(H')$ is a subgroup of $G$
My question is: a homomorphism need not be bijective. Then can we in general talk about its inverse? My guess is that somehow restricting to the inverse image of the subgroup gives us an isomorphism, but cannot see it concretely. Appreciate the insight.