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\title{Group Theory: Sheet 1}
\author{\copyright G. C. Smith 2001}
\date{}
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\textit{The copyright is waived for non-profit making educational
purposes.}

\begin{enumerate}
\item Consider the group $G = \mbox{GL}(n, \mathbb C)$ of $n$ by $n$
invertible matrices with complex entries, the group operation being
matrix multiplication. A subset $S$ of $\mbox{GL}(n, \mathbb C)$
may or may not form a group in its own right (using the same
group operation, and the same identity element $I_n$).
Which of the following subsets of $G$ form groups in their
own right?
\begin{enumerate}
\item[(a)] The elements of $G$ which are symmetric matrices.
\item[(b)] The elements $X \in G$ such that
$X \bar X^T = I_n$. Here $\bar X$ is obtained from $X$ by 
replacing each entry of $X$ by its complex conjugate.
We use superscript $T$ to denote taking the transpose of a matrix.
\item[(c)] The elements $X$ in $G$ which have {\it finite order}. 
(An element $X$ has finite order if there is a positive integer
$m$ such that $X^m$ is the identity element.) 
\end{enumerate}
\item Let $G$ be the symmetric group $\mbox{Sym}(\mathbb Z)$, so
$G$ consist of all bijections from $\mathbb Z$ to $\mathbb Z.$
The {\em order} $n$ of an element $x$ in a group is the smallest
natural number (if there are any) such that $x^n =1$. 
Note that the identity element has
order $1$, and it is possible that there are no natural numbers
with the property -- hence the notion of {\em infinite order}
mentioned above.
\begin{enumerate}
\item Suppose that $p,q$ are positive integers. Prove that there are
elements $\theta, \psi \in G$ with $\theta$ of order $p$,
$\psi$ of order $q$, where $\theta$ commutes with $\psi$.
\item Suppose that $p,q$ are positive integers. Prove that there are
elements $\theta, \psi \in G$ with $\theta$ of order $p$,
$\psi$ of order $q$, but $\theta$ does not commute with $\psi$.
\item Give an example of an element $\zeta \in G$ of infinite order.
\item Give examples of elements $\alpha, \beta \in $, each of
infinite order, such that $\alpha\beta$ has order 2.
\item Suppose that $r$ is a positive integer bigger than 2. 
Do there exist $\mu, \nu \in G$ both of infinite order
such that $\mu \nu$ has order $r$?
\item Do there exist $\sigma, \tau \in G$ both of infinite order
such that $\sigma, \tau$ has infinite order?
\item Suppose that $\eta, \nu \in G$ both have finite order. Does it
follow that $\eta\nu$ has finite order?
\end{enumerate}
\item Let $G$ be a group, and suppose that $x, y \in G$. Prove that
$xy$ and $yx$ have the same order. This means that both have infinite
order, or both have the same finite order.
\item Let $G$ be a group, and suppose that $x, y, z \in G$. 
Prove that $xyz$, $yzx$ and $zxy$ must all have the same order.
Compare this with the order of $xzy$.
\end{enumerate}
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