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Let $G$ and $H$ be groups, and let $X : G\to H$ be a group homomorphism. Prove the following statements.

A. If $K$ is a subgroup of $H$, then $X^{-1} (K)$ is a subgroup of $G$.

B. If $K$ is normal in $H$, then $X^{-1}(K)$ is normal in $G$.

C. If $X$ is surjective and $N$ is a normal subgroup of $G$, then $X (N)$ is a normal subgroup of $H$.

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  • $\begingroup$ I figured out A and B need direction on C $\endgroup$ – Bobby May 1 '13 at 2:28
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We must show $hX(N)h^{-1} \subseteq X(N)$ for all $h \in H$. Let $hxh^{-1} \in hX(N)h^{-1}$. We can find $a \in N, b \in G$ such that $X(a) = x$ and $X(b)= h$. By normality, $bab^{-1} \in X(N)$.

I'll leave the rest to you.

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