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\pagestyle{fancy} \lhead{{\sf Homework 7 $|$ Due October 19 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra I $|$
    Fall 2018 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{0mm}


\begin{enumerate}

\item Prove the lemma from class: Let $N,H,K$ be normal subgroups of
  $G$, with $N\leq H$ and $N\leq K$. If $N$ contains $[G,H]$ and
  $[G,K]$, then it also contains $[G,HK]$. \\



\item Let $G$ be a nilpotent group.
  \begin{enumerate}
  \item Show that every subgroup of $G$ is nilpotent.
  \item Show that every homomorphic image of $G$ is nilpotent. 
  \item Show that nontrivial normal subgroups of $G$ intersect the
    center nontrivially. \\
  \end{enumerate}



\item Let $G$ be a finite group in which every maximal subgroup is normal.
  \begin{enumerate}
  \item Prove that $G$ is nilpotent. [\emph{Hint}: If not, then take a
    non-normal Sylow subgroup $P\leq G$, and choose a maximal $M\leq
    G$ containing $N_G(P)$.]
  \item Show that every maximal subgroup of $G$ has prime index. \\
  \end{enumerate}



\item Let $G$ be a group with $X\subseteq G$, and let $A$ be the
  normal subgroup generated by $X$, i.e.,
  \[
  A=\bigcap\{N\normal G\mid X\subseteq N\}\;.
  \]
  Let $Y=\{gxg^{-1}\mid x\in X,\;g\in G\}$. Show that $A=\<Y\>$. \\



\item If $H$ and $K$ are subgroup of $G$, with $H\normal G$, $H\cap
  K=1$, and $HK=G$, then $G$ is called a semidirect product (or split
  extension) of $H$ by $K$.
  \begin{enumerate}
  \item If $\sigma=(12)\in S_n$, $n\geq 2$, show that $S_n$ is a
    semidirect product of $A_n$ by $\<\sigma\>$.
  \item Show that the dihedral group $D_n$ is a semidirect product of
    two of its non-trivial subgroups. 
  \item Show that the quaternion group cannot be expressed as a
    semidirect product of two non-trivial subgroups. \\
  \end{enumerate}



\item Suppose $H$ and $K$ are groups and $\phi\colon K\to\Aut(H)$ is a
  homomorphism. Let $G$ be the Cartesian product $H\times K$ as a set,
  but with binary operation $(x,y)(u,v)=(x\cdot\phi(y)u,yv)$. 
  \begin{enumerate}
    \item Show that $G$ is a group; denote it by $G=H\rtimes_\phi\,K$,
      and call it the \emph{external semidirect product} of $H$ by $K$
      relative to $\phi$.
    \item Show that $H_1=\{(x,1)\colon x\in H\}\lhd
      G$, $K_1=\{(1,y)\colon y\in K\}\leq G$, and that $G$ is the
      semidirect product of $H_1$ by $K_1$. \\
  \end{enumerate}


\end{enumerate}

\end{document}

