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\title{Group Theory: Sheet 7}
\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 finite abelian group and 
$p$ and prime number which divides
$|G|$. Suppose that $|G| = p^n q$ where $p \not \mid q$.
\begin{enumerate}
\item[(a)] Let $G(p) = \{ g \in G \mid g^{p^n} = 1$. 
Prove that $G(p)$ is a subgroup of $G$.
\item[(b)] Let $G(p') = \{ g \in G \mid g^q = 1  \mbox{ for some } 
q \in \mathbb N \mbox{ with } gcd(p,q) = 1\}.$ 
Prove that $G(p')$ is a subgroup of $G$.
\item[(c)] Prove that $G(p) \cap G(p') = 1$.
\item[(d)] Suppose that $g \in G$ has order $rs$ where $r$ is a power of
$p$ and $s$ is coprime to $p$. Recall that there must exist
$\lambda, \mu \in \mathbb Z$ such that $\lambda r + \mu s =1$.
Prove that $g \in G(p)G(p')$ and deduce that $G =  G(p)G(p')$. 
\item[(e)] Show that $|G(p)| = p^n$ and $|G(p')| = q.$
\item[(f)] Deduce that $G$ must contain an element of order $p$.
\end{enumerate}
\item How many essentially different ways are there to paint the faces
of a cube if two faces must be red, two white and two blue.
{\em Two paintings are essentially the same if you can move 
from one to the other by means of a rigid motion. No reflections are allowed.}
\item You have 24 beads to thread on to a circular string to make
a necklace. There are 8 blue beads, 6 red beads, 4 white beads,
2 black beads and 4 green beads. How many essentially different
necklaces can you make?
\item How many essentially different ways are there to paint half the
vertices of a regular dodecehedron black, and half white?
\item Let $G$ be a finite group. It turns out that the average (over all
group elements)
size of the centralizer of an element must be an integer. Why?
\item Let the group $G$ act on a finite non-empty set 
$\Omega$ of size at least $2$. 
Let $\Gamma$ denote the subset of $\Omega^2$ consisting
of those ordered pairs $(\omega_1, \omega_2)$ where
the entries are distinct. There is a natural action of $G$ on 
$\Gamma$ defined by $(\omega_1, \omega_2) \cdot g
= (\omega_1g, \omega_2g)$ for all $(\omega_1, \omega_2) \in \Gamma$.   
Show that the following statements are equivalent.
\begin{enumerate}
\item[(a)] $G$ has just one orbit as it acts on $\Gamma$.
\item[(b)] $G$ has just one orbit as it acts on
$\Omega$ and there is $\omega \in \Omega$ such that
$G_\omega$ has two orbits as it acts on $\Omega.$
\item[(c)] $G$ has just one orbit as it acts on
$\Omega$ and every $\omega \in \Omega$ has the property
that $G_\omega$ has two orbits as it acts on $\Omega.$
\end{enumerate}
\item Let $G$ be a finite group and $m \in \mathbb N$. 
It turns out that the average (over all group elements)
of the $m$-th power of the size of 
the centralizer of an element must be an integer. Why?
\end{enumerate}
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