Let $\phi : G \to H$ be a group homomorphism and let $A \unlhd G$ and $B \unlhd G$.
- Prove that $\phi^{-1}(B) \unlhd G$.
- Prove that if $\phi$ is surjective, then $\phi(A) \unlhd H$.
- Give an example to show that if $\phi$ is not surjective, then $\phi(A)$ need not be normal in $H$.
Help please, no idea how to start or what to do.
Let $c$ be an element of $\phi^{-1}(B)$ and $g\in G$, we want to prove that $gcg^{-1}$ is in $\phi^{-1}(B)$. Therefore we need to prove that $\phi(gcg^{-1})\in B$ right? This how you start. Now you need to use the hypothesis, $c\in \phi^{-1}(B)$, $\phi$ is a group morphism and $B\triangleleft H$.
For the second question, this is the same idea. Let $c\in \phi(A)$ and $h\in H$, you want to prove $hch^{-1}$ is in $\phi(A)$, that is, there exists $a\in A$ such that $\phi(a)=hch^{-1}$. Now, use the hypothesis you have and the solution will appear naturally.
For the counter example, I suggest that you take $G=A$ and $\phi$ to be some inclusion of $G$ in a bigger group.