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\pagestyle{fancy}
\lhead{MATH 521--01}
\chead{Problem Set \#4\answer{ solutions}}
\rhead{}
\lfoot{}
\cfoot{\answer{Page \thepage\ of \pageref{LastPage}}}
\rfoot{due Tuesday, October 21, 2014}

\begin{document}
\begin{enumerate}
\item \question{Let $G$ be a finite group whose order is greater than
    2 but not divisible by 4. Prove that $G$ contains an element of
    odd order (other than the identity).}
  
  \answer{}
  
\item \question{Let $G$ be an Abelian group of order 15. Prove that
    $G$ is cyclic.}
  
  \answer{}

\item \question{Suppose $|G|=p^2$ with $p$ prime. For non-identity
    elements $a$ and $b$ of $G$, prove that if $\langle
    a\rangle\neq\langle b\rangle$, then every element of the group can
    be expressed in the form $a^kb^\ell$ for integers $k$ and $\ell$.}

  \answer{}

\item \question{Recall that the \emph{center} $Z(G)$ of a group $G$ is
    the subgroup whose elements are of the form $\{x\in G:xa=ax\text{
      for all }a\in G\}$. Prove that $Z(G_1\oplus G_2)=Z(G_1)\oplus
    Z(G_2)$; that is, that the center of a direct product is the
    direct product of centers.}

  \answer{}

\item \question{Prove that $S_3\oplus Z_2$ is isomorphic to $D_6$; you
    may find it helpful to define a specific isomorphism.}

  \answer{}
\end{enumerate}
\end{document}