Group theory - invertible homomorphism and subgroups

Question

I have a question related to the following theorem:

Let $f: G \rightarrow G'$ a be group homomorphism and $K \le G'$. Then $f^{-1}(K) \le G$.

I am assuming (please correct me if I am wrong) that $f^{-1}(K) = \{f^{-1}(k)\; | \; k \in K \}$. The point is, shouldn't we prove that $f^{-1}$ exists, i.e. shouldn't $f$ be an isomorphism? How can the definition of $f^{-1}$ make sense if $f$ is not invertible?

Thanks a lot.

Answer 1

$f^{-1}(K)$ is the preimage of $K$. This is defined for any function $f : A \to B$ between any two sets $A,B$. In fact, for $M \subset B$ $$f^{-1}(M) = \{ a \in A \mid f(a) \in M \} .$$ If you have a group homomorphism $f : G \to G'$ and a subgroup $K$ of $G'$, then it is easy to verify that $f^{-1}(K)$ is a subgroup of $G$.