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\begin{center}
{\large\textbf{Dartmouth College}\\}
Mathematics 101\\\medskip Homework 2 \small{(due Wednesday, Oct 10)}
\end{center}
\begin{enumerate}


\item Two subgroups $H_1$ and $H_2$ of a group $G$ are said to be
  \textit{commensurable}, denoted $H_1 \sim H_2$, if $H_1 \cap H_2$
  has finite index in both $H_1$ and $H_2$.  We want to establish the
  commensurability is an equivalence relation (it is obviously
  reflexive and symmetric).

  \begin{enumerate}
  \item Let $G$ be a group with subgroups, $H$, $K$ each of finite
    index in $G$.  Show that $H\cap K$ has finite index in $G$ by
    establishing the inequality $(G:H\cap K) \le (G:H)(G:K)$.

  \item Show that commensurability is a transitive relation.
  \end{enumerate}


\item Let $G_1$ and $G_2$ be groups with normal subgroups $H_1$ and
  $H_2$.
  \begin{enumerate}
  \item Show that $H_1 \times H_2 \normal G_1 \times G_2$, and that
    $(G_1\times G_2) / (H_1\times H_2) \cong G_1/H_1 \times G_2/ H_2.$
  \item Let $p$ be a prime and $n$ a positive integer.  Determine the
    number of subgroups of $\Z^n$ having index $p$.  Hint: The
    correspondence theorem and last week's homework should be of help.
  \end{enumerate}

\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 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 $G/Z_G \cong Inn(G)$.


  \item Show that if $G/Z_G$ is cyclic, then $G$ is abelian.

  \item Show that if $G$ is a group with cyclic automorphism group,
    then $G$ is abelian.  Hint: In case we haven't covered the
    structure of cyclic groups yet, subgroups of cyclic groups are
    cyclic.
  \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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