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\section*{Math 3140: Homework 9}


\begin{enumerate}
\item[A.]  Find an example of a homomorphism that is neither injective nor surjective.
\item[B.] 
\begin{enumerate}
\item[15.2] Find all normal subgroups of $D_n$.  
\item[15.5]  Find an explicit group $G$ with subgroups $H,K\subseteq G$ such that $H\triangleleft G$, $K\triangleleft H$ and $K\ntriangleleft G$.
\item[15.7]  Let $K\triangleleft G\times H$ be such that 
$$K\cap (\{1_G\}\times H)=\{(1_G,1_H)\}=K\cap (G\times\{1_H\}).$$
Show that $K$ must be abelian.
\item[15.12] Find a proper normal subgroup of $A_4$.  Show that any non-trivial normal subgroup $H$ of $A_5$ must contain a 3-cycle, and use 14.5 to conclude that $H=A_5$, thereby proving $A_5$ is simple.
\item[15.13.]  Suppose $H$ is a cyclic normal subgroup of $G$.  Show that any subgroup of $H$ is also normal in $G$.
\item[(2)]  A group $G$ is \emph{meta-abelian} if there exists and abelian normal subgroup $A\triangleleft G$ such that $G/A$ is also abelian.  Show that $G$ is meta-abelian if and only if $[[G,G],[G,G]]=1$ (the commutator subgroup of the commutator subgroup).
\item[(3)] Find the commutator subgroup of $W_{2,n}$.
\end{enumerate}
\end{enumerate}
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