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\section*{Math 3140: Homework 7}


\begin{enumerate}
\item[A.]  Let $C_r=\langle x\rangle$ be the cyclic group with $r$ elements (but written with multiplication, rather than addition).  Let
$$W_{r,n}=\left\{a\in M_n(C_r\cup\{0\})\ \bigg|\  \begin{array}{l} \text{$a$ has exactly one nonzero entry}\\ \text{in every row and every column}\end{array}\right\}.$$
\begin{enumerate}
\item[(a)] Show that $W_{r,n}$ is a group.   
\item[(b)] Show that $W_{2,2}\cong D_4$.
\item[(c)] What more familiar groups  are $W_{1,n}$ and $W_{r,1}$ isomorphic to?
\item[(d)] What is the order of $W_{r,n}$? 
\end{enumerate}
\item[B.] 
\begin{enumerate}
\item[11.4]  Suppose $|G|$ is the product of two distinct primes.   Show that any proper subgroup of $G$ must be cyclic.
\item[11.7]  Suppose $n\in \ZZ_{\geq 1}$ and $m$ divides $2n$.  Show that $D_{n}$ contains a group of order $m$.
\item[11.8]  Does $A_5$ contain a subgroup of order $m$ for every $m$ that divides $|A_5|=60$?
\item[12.4-5] Find examples of a group $G$ and a subgroup $H$ such that the following sets are \textbf{not} equivalence relations:
\begin{enumerate}
\item[(a)] $\{(x,y)\mid xy\in H\}$,
\item[(b)] $\{(x,y)\mid xyx^{-1}y^{-1}\in H\}$.
\end{enumerate}
\item[12.8]  Let $H$ be a subgroup of a group $G$.
\begin{enumerate}
\item[(a)]  Show that if $|G|=2|H|$, then $gH=Hg$ for all $g\in G$.
\item[(b)]  Show that $gH=Hg$ for all $g\in G$ if and only if $ghg^{-1}\in H$ for all $h\in H$, $g\in G$.
\end{enumerate}
\item[13.2.] Suppose $G$ is abelian with $|G|$ a product of distinct primes.  Show that $G$ is cyclic.
\item[13.4.]  Classify the groups of order $10$.
\end{enumerate}



\end{enumerate}

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