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\begin{center}
  {\large Mathematics 101}\\Fall 2014\\Homework 7
\end{center}

The first couple of problems regard towers of groups.  By a
\textit{normal} tower of groups we mean a chain of subgroups
\[G_0 \normal G_1 \normal \cdots \normal G_r,
\]
that is, where each group $G_k$ is normal in its successor, though not
necessarily normal in $G_r$.

A group $G$ is \textit{solvable} if it admits an \textit{abelian}
tower ending in $\{e\}$, that is if there exist subgroups $G_k \le G$
so that
\[ \{e\} = G_0 \normal G_1 \normal \cdots \normal G_r = G,\]
where each quotient $G_k/G_{k-1}$ is an abelian group, $1\le k \le r$.

A \textit{composition series} for a group $G$ (they always exist for
finite groups) is a normal tower
\[ \{e\} = G_0 \normal G_1 \normal \cdots \normal G_r = G,\]
where each quotient $G_k/G_{k-1}$ is a simple group, $1\le k \le r$,
where simple means having no non-trivial normal subgroups.
\begin{enumerate}
 


\item Let $G$ be a group, and let $G'$ be the \textit{commutator
    subgroup} of $G$, the group generated by the set
  $\{ xyx^{-1}y^{-1} \mid x, y \in G \}$.
  \begin{enumerate}
  \item Show that if $H \le G$, then $H \supseteq G'$ if and only if
    $H \normal G$, and $G/H$ is an abelian group. In particular
    $G' \normal G$ is the smallest normal subgroup so that $G/G'$ is
    abelian.
  \item Show that any group homomorphism $\varphi: G \to H$ where $H$
    is abelian factors through the quotient $G/G'$.
  \item Given a group $G$, define a sequence of subgroups $G^{(k)}$ by
    $G^{(1)}= G'$, the commutator subgroup, and
    $G^{(k+1)} = [G^{(k)}]'$, the commutator of $G^{(k)}$.  Show that
    $G$ is solvable if and only if $G^{(k)} = \{e\}$ for some
    $k \ge 1$.
  \end{enumerate}

\item Show that the following three statements concerning finite
  groups are equivalent.  The second is the famous Feit-Thompson
  theorem.

  \begin{enumerate}
  \item A finite non-abelian simple group has even order.
  \item A simple group of odd order is isomorphic to $\Z/p\Z$ for some
    prime $p$.
  \item Every group of odd order is solvable.
  \end{enumerate}

\item Let $G$ be a finite group and $H \normal G$.  Show that $G$ has
  a composition series one of whose terms is $H$.

  \vfill \rightline{(continued)}
  \newpage
\item Group Actions.
  \begin{enumerate}
  \item Let $G$ be a finite group, and $H$ a proper subgroup.  Show
    that $G$ is not the union of conjugates of $H$.
  \item Let $G = GL_n(\C)$, and $H$ the subgroup of lower triangular
    matrices.  Show that $G$ is the union of conjugates of $H$.
  \item Let $G = GL_n(\R)$, and $H$ the subgroup of lower triangular
    matrices.  Determine whether or not $G$ is the union of conjugates
    of $H$ (proof or counterexample).
  \end{enumerate}

\item Let $G$ be a non-abelian group of order $p^3$, $p$ a prime.
  Show that the commutator of $G$ equals its center.

\item Let $p < q$ be primes, and $G$ a non-abelian group of order
  $pq$.  Show that there is an embedding of $G$ into the symmetric
  group $S_q$.
\end{enumerate}






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