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\begin{document}

\textbf{Math 6310, Assignment 2 \hfill Due in class: Friday, September 25}

% \medskip {\large Name: }

\begin{enumerate}

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\item Let $x,y\neq e$ be elements of a group such that $xyx^{-1}=y^5$, the element $x$ has order $3$, and $y$ has odd order. Find the order of $y$. 

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\item Let $V$ be the set of $r \times s$ matrices over $\RR$. Consider the group $G=\GL_r(\RR)\times\GL_s(\RR)$ acting on $V$, where
\[
(A,B)\colon M\mapsto AMB^{-1}\quad\text{ for } M\in V \text{ and } (A,B)\in G.
\]
How many orbits are there? Describe them.

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\item Let $G$ be a group, and $Z$ its center.
\begin{enumerate}
\item If $G/Z$ is cyclic, prove that $G$ is abelian.
\item If $\Aut(G)$ is cyclic, prove that $G$ is abelian.
\item If $|G|=p^3$ for $p$ a prime, show that either $G$ is abelian or $|Z|=p$.
\end{enumerate}

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\item Let $G$ be a $p$-group. If $H$ is a normal subgroup of order $p$, prove that $H$ is contained in the center of $G$.

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\item Let $G$ be a finite group with an automorphism $\phi$ that fixes only the group identity.
\begin{enumerate}
\item Show that each element of $G$ can be written as $x^{-1}\varphi(x)$.
\item If $\phi^2$ is the identity, prove that $\phi(x)=x^{-1}$ for each $x$, and conclude that $G$ is abelian.
\end{enumerate}

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\item Let $G$ be a finite group such that $\Aut(G)$ acts transitively on the set $G\smallsetminus\{e\}$. Show that $G$ is a $p$-group, and that~$G$ is abelian.

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\item Let $G$ be an infinite group, and $H$ a subgroup of finite index. Show that $G$ has a normal subgroup $K$ of finite index, with $K < H$.

%\textbf{Solution:} Consider the action of $G$ on $G/H$ by left translation, $\phi\colon G\to\Perm(G/H)$.

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\item Let $G$ be an infinite group with an element $x\neq e$ that has only finitely many conjugates. Prove that $G$ is not simple, i.e., that it has a proper normal subgroup.

%\textbf{Solution:} Consider $G$ acting on itself by conjugation.

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\item Let $H$ and $K$ be subgroups of a group $G$. For $x\in G$, the set $HxK = \{hxk\mid h\in H, k\in K\}$ is a \emph{double coset}.
\begin{enumerate}
\item Prove that $G$ is a disjoint union of double cosets $HxK$, and that
$\displaystyle{|HxK| = \frac{|H||K|}{|H\cap xKx^{-1}|}}$.
\item Prove that $\displaystyle{|HK| = \frac{|H||K|}{|H\cap K|}}$.
\item If all double cosets $HxH$ for $x\in G$ have the same number of elements, show that $H\nor G$.
\end{enumerate}

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\item Let $G$ be a group of odd order acting transitively on a set $S$. Fix $s\in S$. Show that the orbits of the action of $G_s$ on $S\setminus\{s\}$ have lengths that are equal in pairs.

%\textbf{Solution:} As the action is transitive, we may take $S=G/G_s$; the problem reduces to $G_s$ acting on $\{xG_s\mid x\notin G_s\}$ by left translation. We claim the orbits of $xG_s$ and $x^{-1}G_s$ under this action are disjoint and have the same length.

\end{enumerate}
\end{document}

