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\title{Group Theory: Math30038, Sheet 7}
\date{GCS}
\author{}
\begin{document}
\maketitle

\begin{enumerate}
\item Suppose that $n \geq 3$. Let $g \in G = S_n$ be an $n$-cycle.
Recall that all $n$-cycles are conjugate in $S_n$.
\begin{enumerate}
\item[(a)] Prove that $C_G(g) = \langle g \rangle.$
\item[(b)] Deduce that $Z(G) = 1$.
\item[(c)] Show that $\mbox{Aut}(G)$ has a subgroup which 
is isomorphic to $G$.
\end{enumerate}
\item Suppose that $G$ is a group and that $\mbox{Aut}(G)$ is the trivial group.
\begin{enumerate}
\item[(a)] Show that $G$ must be abelian.
\item[(b)] By considering the inversion map, deduce that $g^2=1$ for all $g \in G$.
\item[(c)] Write $G$ additively, so that $G$ becomes a vector space over the field
$\mathbb Z_2$ of ``integers modulo 2''. You may assume that every vector space has a basis.
Deduce that $G$ must either be cyclic of order 2 or the trivial group.
\end{enumerate}
\item Suppose that $G$ is a group and that $\mbox{Aut}(G)$ is finite.
Prove that $|G:Z(G)|$ must be finite.
\item (Challenge) Suppose that $G$ is a group and that $\mbox{Aut}(G)$ is 
cyclic. Prove that $G$ must be abelian.
\item Prove that for a finite group, the notions of isomorphism, monomorphism
and epimorphism all co-incide.
\item In each case give an example of a group $G$ and a map $\zeta: G 
\rightarrow G$ which satisfies the specified property.
\begin{enumerate}
\item[(a)] An epimorphism (i.e. a surjective homomorphism) but
not an isomorphism (suggestion for $G$: fix a prime number $p$ and
consider those complex numbers $z$ such that $z^{p^n} =1$ for some 
natural number $n$ which may vary as $z$ varies).
\item[(b)] A monomorphism (i.e. an injective homomorphism)  but
not an isomorphism (suggestion for $G$: $\mathbb Z$ under addition).
\end{enumerate}   
\item Suppose that $G$ is a finitely generated group (i.e. there is
a finite subset $X$ of $G$ such that $\langle X \rangle = G$) and that
every conjugacy class of $G$ is finite. Prove that $|G:Z(G)|$ is finite.
\item Let $V$ be a vector space of dimension $n$ over a field $F$ which
has $q$ elements. Let $\mbox{GL}(V)$ denote the group of linear
isomorphisms from $V$ to $V$ (a group under composition of maps).
Determine $|\mbox{GL}(V)|$.
\item 
\begin{enumerate}
\item[(a)] Suppose that $G$ is a group acting on a set $\Omega$.
Define a map $\eta: G \longrightarrow \mbox{Sym}(\Omega)$
by $(g)\eta : \omega \mapsto \omega \cdot g$ for all $\omega \in
\Omega$ and for all $g \in G$. Prove that $\eta$ is a 
homomorphism of groups.
\item[(b)] Suppose that $\eta: G \longrightarrow \mbox{Sym}(\Omega)$
is a group homomorphism. Show that one may define a group action of
$G$ on the set $\Omega$ by defining $\omega \cdot g$ to
be $(\omega)((g)\eta)$ for all $\omega \in \Omega$ and for all
$g \in G$. Show that this is indeed a group action.
\item[(c)] Show that the procedures outlined in parts (a) and (b)
are mutually inverse (i.e. if you perform one, then the other, then
you recover the situation in which you started).
\end{enumerate} 
\end{enumerate}  
\end{document}
