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\begin{center}
{\large\textbf{Dartmouth College}\\}
Mathematics 101\\\medskip Homework 2 \small{(due Thursday, Oct 8)}
\end{center}
\begin{enumerate}
\item Let $G$ be a group and $H \le G$.  Recall that the normalizer of
  $H$ in $G$ is defined by $N_G(H) = \{g \in G \mid gHg^{-1} = H\}$.
  Of course if $H$ is a finite group, this condition can be weakened
  to $gHg^{-1} \subseteq H$, however this is not true in general.  In
  particular let $G = GL_2(\Q)$ and $H$ the following group of
  unipotent matrices in $SL_2(\Z)$: $H = \{\spmatrix{1}{m}{0}{1} \mid
  m \in \Z\}$.  

  \begin{enumerate}
  \item Characterize the diagonal matrices $g \in GL_2(\Q) \cap M_2(\Z)$
 with $gHg^{-1} \subseteq H$.
\item  Show that there exist $g$ among those just found which do not
  live in the normalizer.
  \end{enumerate}


\item  Let $G$ be a cyclic group of order $n$, and let $d \in \Z_+$
  with $d\mid n$.  Show that $G$ has exactly $d$ elements of exponent
  $d$.

\item  Let $G$ be a group with subgroups, $H$, $K$ each of finite
  index in $G$, say $[G:H] = m$ and $[G:K] = n$.  Show that $H\cap K$ has finite index in $G$.  More
  specifically show that
\[
\lcm\{m,n\} \le   [G:H\cap K] \le mn.
\]

\item  For a group $G$, the \textit{center} of $G$, denoted $Z_G$ is
  defined by: \newline
 $Z_G = \{ x \in G \mid xg = gx \textrm{ for all } g \in G\}$.
Show that if $G/Z_G$ is cyclic, then $G$ is abelian.

\item  Let $Aut(G)$ denote the group  of automorphisms of a group $G$,
  and $Inn(G)$ the subgroup of inner automorphisms.  That is,
  $Inn(G) = \{ \varphi_g \mid g \in G\}$ where $\varphi_g: G \to G$ is
  defined by $\varphi_g(x) = g x g^{-1}$.
  \begin{enumerate}
  \item Show that $Inn(G) \trianglelefteq Aut(G)$.
    \item Show that $G/Z_G \cong Inn(G)$.
      \item Show that if $Aut(G)$ is cyclic, then $G$ is abelian.
  \end{enumerate}

\item  Let $G$ be a finite group, and assume $N$ is a normal subgroup
  of $G$ with \newline
$\gcd(|N|, [G:N]) = 1$.  Show that $N$ is the only
  subgroup of $G$ of order $|N|$.


\end{enumerate}


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