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\section*{Math 3140: Homework 6}


\begin{enumerate}
\item[A.] 
\begin{enumerate}
\item[9.1] Do either of the following sets of $n\times n$ matrices form a group under matrix multiplication?
\begin{enumerate}
\item[(a)] Diagonal matrices, $\{a\in M_n(\RR)\ \mid\ a_{ij}=0, i\neq j, a_{ii}\neq 0\}.$
\item[(b)] Symmetric matrices, $\{a\in M_n(\RR)\ \mid\ a_{ij}=a_{ji}, 1\leq i,j\leq n\}.$
\end{enumerate}
\item[9.3] Prove that the elements of $\GL_n(\RR)$ which have integer entries and determinant in $\{1,-1\}$ form a subgroup of $\GL_n(\RR)$. Note that this group is in fact $\GL_n(\ZZ)$. Hint: Recall Cramer's Rule from linear algebra.
\item[10.1] Show that if $G\times H$ is cyclic, then both $G$ and $H$ are cyclic.
\item[10.2] Show that $\ZZ\times\ZZ$ and $\ZZ$ are not isomorphic.
\item[10.7]  Which of the following groups are isomorphic to one-another?  
$$\ZZ_{24},\quad D_4\times \ZZ_3,\quad D_{12},\quad A_4\times \ZZ_2,\quad \ZZ_2\times D_6,\quad S_4,\quad \ZZ_{12}\times\ZZ_2.$$
\end{enumerate}
\item[B.]  For $p$ prime, let $\FF_p$ denote the set $\{0,1,\ldots, p-1\}$ where we add \textbf{and} multiply modulo $p$ (as opposed to $\ZZ_p$ where we just add).  Define
$$U_n(\FF_p)=\{a\in M_n(\FF_p)\ \mid\ a_{jj}=1, 1\leq j\leq n, a_{ji}=0, 1\leq i<j\leq n\}.$$
This group is called the \emph{group of unipotent, uppertriangular matrices with entries in $\FF_p$.}
\begin{enumerate}
\item[(a)]  What is the order of $U_3(\FF_2)$?  Show that $U_3(\FF_2)$ is isomorphic to an already familiar group. \\ \textbf{Remark.} The group $U_3(\FF_p)$ is often called the \emph{Heisenberg group} and is useful in mathematical physics.
\item[(b)] Show that $U_2(\FF_p)\cong \ZZ_p$, and that if $n\geq 2$, then $U_2(\FF_p)$ is isomorphic to a subgroup of $U_n(\FF_p)$.
\end{enumerate} 
\end{enumerate}

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