\documentclass{article}
\usepackage{amssymb}
\usepackage{latexsym}
\title{Group Theory: Math0038, Sheet 1}
\author{GCS}
\date{}
\begin{document}
\maketitle

{\it The course web site is available via
{\tt http://www.bath.ac.uk/$\sim$masgcs/}}

\begin{enumerate}
\item {\em Let $G$ be a group. Suppose that $x \in G$.
Let \[ C_G(x) = \{ g \in G \mid  gx = xg \} \subseteq G.\]
Prove that $C_G(x) \leq G$.} \newline
\noindent {\bf Solution: } $1 \in C_G(x) \not = \emptyset$. Moreover if
$h, g \in C_G(x)$, then $hx = xh$ and $gx = xg$ so
$xg^{-1} = g^{-1}x$ and therefore $hg^{-1}x = 
hxg^{-1} = xhg^{-1}.$ Thus $hg^{-1} \in C_G(x)$ and therefore
$C_G(x) \leq G$.
\item {\em Let $G$ be a group. Suppose that $S \subseteq G$.
Let \[C_G(S)  = \{ g \in G \mid  gs = sg \forall s \in S\}.\]
Prove that $C_G(S) \leq G$.}  \newline
\noindent {\bf Solution: } $C_G(S) = \cap_{s \in S} C_G(s)$
and the intersection of subgroups is a subgroup.
\item {\em Let $G$ be a group. Suppose that $S \subseteq G$.
Let \[N_G(S) = \{ g \in G \mid gS = Sg \}\] where
$gS = \{ gs \mid s \in S\}$ and $Sg = \{ sg \mid s \in S \}$.
Prove that $C_G(S) \leq N_G(S)  \leq G$.}  \newline
\noindent {\bf Solution: } It is routine to prove that
$N_G(S)$ is a subgroup of $G$ (see previous solution for the
outline).
\item {\em Let $G$ be a group. Suppose that $S \subseteq G$
and that $x \in N_G(S)$. Prove that
$C_G(S)x = xC_G(S)$.} \newline
\noindent {\bf Solution: } We will show that $x^{-1}C_G(S)x
= C_G(S)$ whenever ****.  
\item Let $G$ be a finite group. Suppose that
$\emptyset \not = H \subseteq G$ has the property that
if $a,b \in H$, then $ab \in H$. Does it follow that $
H  \leq G$? What happens if we relax the condition that
$G$ is finite?
\item Suppose that $G$ is a group and that $x \in G$.
Prove that $(x^{-1})^{-1} = x$.
\item Does there exist a group $G$ containing 
elements $a,b$ such that $a^2 = b^2 = (ab)^3 = 1$?
\item Suppose that $G$ is a group with the property that
$x^2 = 1$ whenever $x$ is an element of $G$. Show that
$G$ must be abelian. 
\item (Challenge) Suppose that $G$ is a group with the property that
$x^3 = 1$ whenever $x$ is an element of $G$. Show that
$G$ need not be abelian. 
\end{enumerate}
\vfill
\end{document}
\documentclass{article}
\usepackage{amssymb}
\usepackage{latexsym}
\title{Group Theory: Math0038, Sheet 1}
\author{GCS}
\date{}
\begin{document}
\maketitle

{\it The course web site is available via
{\tt http://www.bath.ac.uk/$\sim$masgcs/}}

\begin{enumerate}
\item Let $G$ be a group. Suppose that $x \in G$.
Let \[ C_G(x) = \{ g \in G \mid  gx = xg \} \subseteq G.\]
Prove that $C_G(x) \leq G$.
\item Let $G$ be a group. Suppose that $S \subseteq G$.
Let \[C_G(S)  = \{ g \in G \mid  gs = sg \forall s \in S\}.\]
Prove that $C_G(S) \leq G$.
\item Let $G$ be a group. Suppose that $S \subseteq G$.
Let \[N_G(S) = \{ g \in G \mid gS = Sg \}\] where
$gS = \{ gs \mid s \in S\}$ and $Sg = \{ sg \mid s \in S \}$.
Prove that $C_G(S) \leq N_G(S)  \leq G$.
\item Let $G$ be a group. Suppose that $S \subseteq G$
and that $x \in N_G(S)$. Prove that
$C_G(S)x = xC_G(S)$.
\item Let $G$ be a finite group. Suppose that
$\emptyset \not = H \subseteq G$ has the property that
if $a,b \in H$, then $ab \in H$. Does it follow that $
H  \leq G$? What happens if we relax the condition that
$G$ is finite?
\item Suppose that $G$ is a group and that $x \in G$.
Prove that $(x^{-1})^{-1} = x$.
\item Does there exist a group $G$ containing 
elements $a,b$ such that $a^2 = b^2 = (ab)^3 = 1$?
\item Suppose that $G$ is a group with the property that
$x^2 = 1$ whenever $x$ is an element of $G$. Show that
$G$ must be abelian. 
\item (Challenge) Suppose that $G$ is a group with the property that
$x^3 = 1$ whenever $x$ is an element of $G$. Show that
$G$ need not be abelian. 
\end{enumerate}
\vfill
\end{document}

A
A
A
A
A
 
