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\title{Group Theory: Sheet 5}
\author{\copyright G. C. Smith 2001}
\date{}
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\textit{The copyright is waived for non-profit making educational
purposes.}

{\it The course web site is available via 
{\tt http://www.bath.ac.uk/$\sim$masgcs/}}

\begin{enumerate}
\item Let $G$ be a group, and let $S$ be a subset of $G$.
Let \[ C_G(S) = \{ g \in G \mid g^{-1}sg = s \forall s \in G\} \mbox{ and }
N_G(S) = \{ g \in G \mid g^{-1}Sg = S\}.\]
\begin{enumerate}
\item[(a)] Prove that $C_G(S) \leq N_G(S) \leq G$.
\item[(b)] Prove that $C_G(S) \unlhd N_G(S)$.
\item[(c)] Show that $N_G(S)/C_G(S)$ is isomorphic to  
a subgroup of $\mbox{Sym} S$.
\end{enumerate}

\item Let $G$ be a finitely generated group in which for each $x \in G$, the
set $S_x = \{g^{-1}xg \mid g \in G\}$ is finite. Let $Z(G)$ denote
the centre of $G$ (the subgroup of $G$ consisting of those elements
of $G$ which commute with all elements of $G$). Prove that
$Z(G)$ has finite index in $G$. {\em (Hint: Do Question 1 first)}

\item Prove that $\mbox{Aut }S_5$ is isomorphic to $S_5$.

\item Suppose that $G = \langle x, y \rangle$ and $x^2 = y^2 = 1$.
Prove that $G$ has a cyclic subgroup of index at most 2.

\item Suppose that $G = \langle S \rangle$ is a group and that
$S$ is finite. Now suppose that also $G = \langle T \rangle$.
Show that there is a finite subset $U$ of $T$ such that
$G = \langle U \rangle$.

\item Consider the maps $\alpha, \beta \in \mbox{Sym }\mathbb Z$
defines by $\alpha: i \mapsto i+2$ for all $i \in \mathbb Z$, and
$\beta = (0,1)$. Thus $\beta$ fixes all integers except $0$ and $1$,
and it swaps those. Consider the groups
$G = \langle \alpha, \beta \rangle$ and 
$H = \langle T \rangle$ where $T = \{ 
\alpha^{-k} \beta \alpha^{k} \mid k \in \mathbb Z\}$. Observe that
all elements of $H$ have finite support (i.e. each one moves only
finitely many integers). Deduce
that a finitely generated group may have a subgroup which has
no finite generating set.

\item Let $G$ be a finite cyclic group of order $n$. Suppose that
$d$ is a divisor of $n$. Show that $G$ has a unique subgroup of order 
$d$.
\end{enumerate}
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