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{\large\textbf{Dartmouth College}\\}
Mathematics 101\\\medskip Homework 3 \small{(due Thursday, Oct 15)}
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\begin{enumerate}



\item Show that the following three statements about finite groups are
  equivalent. The second is the 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 $\Z/p\Z$ where
      $p$ is a prime.
      \item Every group of odd order is solvable.
  \end{enumerate}

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

\item  Let $G$ be a group and let $G'$ be the subgroup of $G$ generated
  by the set \newline $\{xyx^{-1}y^{-1} \mid x,y \in G\}$.  $G'$ is
  called the commutator subgroup of $G$.
  \begin{enumerate}
  \item Show that if $H$ is a subgroup of $G$, then $H \supseteq G'$
    if and only if $H \triangleleft G$ and $G/H$ is abelian.  In
    particular, $G'\triangleleft G$ and $G/G'$ is abelian.

  \item Show that if $\varphi: G \to H$ is a homomorphism of groups,
    and $H$ is abelian, then $\varphi$ factors through $G/G'$, that is
    there is a map $\varphi_* : G/G' \to H$ with $\varphi =
    \varphi_* \circ \pi$ with $\pi: G \to G/G'$ the standard
    projection.

 


  \end{enumerate}

 \item For 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 $k \ge 1$.

 \item Let $G$ be a group, $H$ and $K$ solvable subgroups of $G$, with
   $K \triangleleft G$.  Show that $HK$ is a solvable subgroup of $G$.

\end{enumerate}


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