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\title{Group Theory: Sheet 4}
\author{\copyright G. C. Smith 2001}
\date{}
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\maketitle


\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.
\begin{enumerate}
\item[(a)] Suppose that $H \leq G$ and $|G:H| = 2$. Prove that
$H \unlhd G$.
\item[(b)] Show that $S_3$ has a non-normal subgroup of index 3.
\end{enumerate}

\item Let $G$ be a group, and suppose that $H, K$ are subgroups
of $G$.
\begin{enumerate}
\item[(a)] Show that $HK$ is a subgroup of $G$ if and only if $HK
= KH$.
\item[(b)] Suppose that $K \unlhd G$. Show that $HK$ is a subgroup
of $G$.
\item[(c)] Suppose that $G$ is a finite group. Show that
$|HK| \cdot |H \cap K| = |H| \cdot |K|.$
\end{enumerate}

\item Suppose that $G$ is a group and that $N$ is a finite
normal subgoup of $G$. Show that $G$ must have a subgroup $H$
of finite index with the property that every element of $H$ commutes
with every element of $N$. {\em Hint: for each $g \in G$ there
$\sigma_g \in \mbox{Aut}(N)$ defined by $n \mapsto g^{-1}ng$
for every $n \in N$.}

\item Let $G$ be a group. 
\begin{enumerate}
\item[(a)] Show that $x, y \in G$ commute
if and only if $x^{-1}y^{-1}xy= 1.$
\item[(b)] Suppose that $M, N$ are normal subgroups of $G$
such that $M \cap N = 1$. Show that each element of $M$ commutes
with each element of $N$.
\item[(c)] Suppose that $M, N$ are normal subgroups of $G$
such that both $G/M$ and $G/N$ are abelian. Prove that $G$
is an abelian group.
\end{enumerate}

\item Show that $\mbox{Aut}(S_4)$ is isomorphic to $S_4$.

\item
\begin{enumerate}
\item Suppose that $G$ is a group.  
Show that the map $i: G \rightarrow G$
defined by $g \mapsto g^{-1}$ for every $g \in G$ is an 
automorphism of $G$ if and only if $G$ is abelian.
\item Let $G$ be a group with the property that $g^2 = 1$
for every $g \in G$. Prove that $G$ must be abelian, and 
show that $G$ (written additively) may be regarded as a vector
space over the field $\mathbb F_2$ with two elements.
\item Suppose that $G$ is a finite group with a 
trivial automorphism group.
Prove that $G$ has either 1 or 2 elements.
\end{enumerate}
\end{enumerate}
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