\documentclass{article} \usepackage{amssymb} \usepackage{latexsym} \title{Group Theory: Sheet 7} \author{\copyright G. C. Smith 2001} \date{} \begin{document} \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 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} \end{document}