\documentclass{article} \usepackage{amssymb} \usepackage{latexsym} \title{Group Theory: Extra Exercises} \author{} \begin{document} \begin{enumerate} \item Suppose that $G$ is a group. \begin{enumerate} \item[(a)] Define a relation $\sim$ on $G$ by $x \sim y$ if and only if there exists $g \in G$ such that $g^{-1} x g = y.$ Prove that this is an equivalence relation on $G$ (see section 1.18 in the book). The equivalence classes of this equivalence relation are called the conjugacy classes of $G.$ \item[(b)] Prove that $G$ is an abelian group if and only if each of its conjugacy classes has size 1. \item[(c)] Calculate the conjugacy classes of $S_3,$ the symmetric group on three letters (see section 5.3 in the book). \item[(d)] Give an example of a group which has infinitely many distinct conjugacy classes. \end{enumerate} \item Suppose that $G$ is a group. A subgroup $N$ of $G$ is said to be {\em normal} in $G$ if and only if whenever $x \in G,$ then $xN = Nx.$ \begin{enumerate} \item[(a)] Show that the group $S_3$ has a normal subgroup of size $3.$ \item[(b)] Prove that the subgroup $M$ of $G$ is normal in $G$ if and only if $x^{-1}Nx = N$ for every $x \in G.$ \item[(c)] Suppose that $N$ is a normal subgroup of $G,$ that $C$ is a conjugacy class of $G$ (see question 1), and that $N \cap C \not = \emptyset.$ Prove that $C \subseteq N.$ \end{enumerate} \item Suppose that $G$ is a group, that $x \in G$ and that $C$ is the conjugacy class of $G$ which contains $x.$ Let $C_G(x) = \{ g \mid g \in G,\ g^{-1}xg = x\}$ (this set is called the {\em centralizer of $x$ in $G$}). \begin{enumerate} \item[(a)] Show that $C_G(x)$ is a subgroup of $G.$ \item[(b)] Define Let $S = \{ C_G(x)a \mid a \in G\}$ be the set of right cosets of $C_G(x)$ in $G.$ Attempt to define a map $\theta : S \rightarrow C$ by $C_G(x)z \mapsto z^{-1}xz.$ {\em You need to be careful here. You need to verify that if $C_G(x)z_1 = C_G(x)z_2,$ then $z_1^{-1}xz_1 = z_2^{-1}xz_2,$ otherwise this definition makes no sense. Think about this remark carefully, and make sure you understand its significance. This is a subtle and important point.} \item[(c)] Show that the map $\theta$ of part (b) is bijective. \item[(d)] Show that if $G$ is a finite group, then $|C| |C_G(x)| = |G|.$ \end{enumerate} \end{enumerate} \end{document}