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\begin{center}
  \textbf{Math 71}\\ Homework 4
\end{center}



\begin{itemize}

\item[p89. \#36] Prove that if $G/Z(G)$ is cyclic, then $G$ is
  abelian.  Here $Z(G)$ is the center of $G$.  Hint: If $G/Z(G)$ is
  cyclic with generator $xZ(G)$, show that every element of $G$ can be
  written as $x^az$ for some integer $a$ and some element
  $z \in Z(G)$.

\item [p89: \#41] Let $G$ be a group.  Prove that
  $N = \la x^{-1}y^{-1}xy\mid x,y \in G\}$ is a normal subgroup of $G$
  and $G/N$ is abelian.  $N$ is called the commutator subgroup (see
  problem 40).

\item [p96: \#11] Let $H \le K \le G$.  Prove that
  $|G:H| = |G:K| |K:H|$.  Do not assume that $G$ is finite.

\item [p96: \#18] Let $G$ be a finite group, let $H$ be a subgroup of
  $G$ and let $N \normal G$. Prove that if $|H|$ and $|G:N|$ are
  relatively prime, then $H \le N$.

\item [p96: \#19] Prove that if $N$ is a normal subgroup of a group
  $G$, and $\gcd(|N|, |G:N|) = 1$, then $N$ is the unique subgroup of
  $G$ having order $|N|$.

\item [p101: \#3] Prove that if $H$ is a normal subgroup of $G$ having
  prime index $p$, then for all $K \le G$ either
  \begin{itemize}
  \item $K \le H$ or
  \item $G = HK$ and $|K:K\cap H| = p$.
  \end{itemize}
\end{itemize}

 

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