Hi: Let G be a group and N normal in G such that 1 and N are the only normal subgroups of G that are contained in N. Let $\phi$ be a surjective (onto) homomorphism from G to a group H. Let $A \neq 1$ be a normal subgroup of $H$ that is contained in $N^\phi$ and let $M = A^{\phi^{-1}} \cap N$. Then $M$ is normal in $G$.
OK. But I am said that, then,$M \neq 1$ since $A \neq 1$. I have been trying hard to prove that $M neq 1$ but I could not. Any suggestion?