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\title{Group Theory: Math30038, Sheet 6}
\date{GCS}
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\maketitle

\begin{enumerate}

\item Consider the group $D$ of rigid symmetries of a regular
$n$-gon (which may be turned over). Prove that this group has
order $2n$, is non-abelian, and can be generated by two elements 
each of order 2. Show that $D$ has a cyclic subgroup of
index 2.

\item Consider the group $D$ of rigid symmetries of the integers:
so $D$ is the group of all bijections $\theta$ from $\mathbb Z$ to $\mathbb Z$ 
which preserve distance. Thus $\theta$ must have the property that
if $x, y \in \mathbb Z$, then $|x-y| = |(x)\theta - (y)\theta|.$
Prove that this group has infinite order, 
is non-abelian, and can be generated by two elements
each of order 2. Show that $D$ has a cyclic subgroup of
index 2.

\item Let $D = \langle x, y \rangle$ where $o(x) = o(y) = 2$
and $x \not = y$. Let $z = xy$ and put $H = \langle z \rangle$.
\begin{enumerate}
\item Prove that $x^{-1}zx = y^{-1}zy = z^{-1}.$
\item Prove that $x, y  \not \in H$.
\item Prove that $|G:H| = 2$.
\item Let $n = o(z) \in \mathbb N \cup \{ \infty\}.$ For each
possible value of $n$ let $G$ be called $D_n$. Show that the
multiplication in $D_n$ is completely determined (i.e. the number
$n$ nails down the group).
\item For each $n \in \mathbb N \cup \{ \infty\}$, determine the
centre of $D_n$.
\item Determine the conjugacy classes of $D_8$.
\item Do you recognize $D_6$?
\end{enumerate}

\item Suppose that $G$ is a non-abelian finite group of
order $2p$ where $p$ is a prime number. Prove that
$G$ is generated by two elements 
order 2.

\item We define a subgroup $Q$ of $S_8$ by letting 
$i = (1,2,3,4)(5,6,7,8)$, $j = (1,5,3,7)(2,8,4,6)$ 
({\bf and NOT $(2,6,4,8)$ as earlier stated})
and put $Q = \langle i, j \rangle$. Let $k = ij$
and $z = i^2$. {\em This group was the basis of William Rowan Hamilton's
generalization of the complex numbers called the
{\rm Quaternions.}}
\begin{enumerate}
\item Show that $i^2 = j^2 = k^2  = z$ and $z^2 = 1$.
\item Show that $ij = k$, $jk = i$ and $ki = j$.
\item Show that $ji = zk$, $kj = zi$ and $ik = zj$.
\item Show that $z$ is in the centre of $Q$.
\item Show that $Q = \langle i \rangle \cup z \langle i \rangle$.
\item Show that $|Q| = 8$.
\item Show that $Q$ and $D_8$ (in Question 1) are both non-abelian
groups of order 8, but they contain different numbers of elements of order
4.
\item Determine the conjugacy classes of $Q$.
\item On which bridge are the quaternions inscribed?
\end{enumerate}

\item Let $G$ denote the set of invertible $n$ by $n$ matrices
with complex entries. This is a group under multiplication of
matrices. Give a transversal for the conjugacy
classes of $G$. {\em Hint: the course MA20012 does this (and not much else).}

\item Show that if $N$ is a normal subgroup of $G$, then $N$ must be
a union of conjugacy classes of $G$. Deduce that the only normal subgroups
of $A_5$ are $1$ and $A_5$, but that $A_4$ has a normal subgroup $M$ which
is neither 1 nor $A_4$.

\end{enumerate}
\end{document}
