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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 10.2  \\ Spring 2019 \hfill Due April 10}}}

\large

\vspace{1cm}


\begin{exercise}
Let $G$ be a group and let $G'$ denote its commutator subgroup.  Set $G^{(1)} = G'$ and recursively define $G^{(i+1)}=(G^{(i)})'$.  Suppose we have descending a sequence of subgroups
\begin{equation}\label{series}
 \cdots \lhd G_2 \lhd G_1 \lhd G_0 = G
\end{equation}
so that $G_i/G_{i+1}$ is abelian for all $i$.  Prove that $G^{(i)} < G_i$ for all $i$.  [{\em Suggestion:}  Induct.  Use the fact that, given $H \lhd G$, $G/H$ is abelian if and only if $G'<H$.]
\end{exercise}

\begin{exercise}  
Recall that $G$ is {\em solvable} provided $G^{(r)} = \{ e \}$ for some $r \ge 1$.  Prove that $G$ is solvable if and only if there is a sequence \eqref{series}, with abelian factors, that eventually terminates with the trivial group.  [{\em Suggestion:}  For one direction, use the preceding exercise.]
\end{exercise}

\begin{exercise}
Let $G$ be a finite group with subgroups $H$ and $N$.  If $N$ is normal in $G$, prove that $[H:H \cap N]$ divides $[G:N]$.  [{\em Suggestion:}  This follows almost immediately from the Second Isomorphism Theorem and the Correspondence Principle.  Alternatively, one can use the First Isomorphism Theorem to find a monomorphism $H/(H \cap N) \hookrightarrow G/N$.]
\end{exercise}

\begin{exercise}
Let $H$ be a finite group of odd order.  Given $x \in H$, let $T_x \in \operatorname{Perm}(H)$ denote left translation by $x$.  Prove that $T_x$ is even.  [{\em Suggestion:}  Recall that the mapping $x \mapsto T_x$ is a monomorphism $H \hookrightarrow \operatorname{Perm}(H) \cong S_n$, where $n = |H|$.  Apply the preceding exercise with $G = S_n$ and $N = A_n$.  It's also possible to show that every cycle of $T_x$ has odd length.]
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