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\title{General Mathematics and CPS II}
\author{Exercise 9}
\date{March 4, 2015}
\maketitle

\begin{enumerate}

\item Let $G$ be a finite group (i.e., a group with a finite number of
elements), and let $a \in G$.  Show that there exists some $n \in
\mathbb N$ such that $a^n=e$.  

\emph{Recall:} $a^n$ is understood as letting the group operation act
between $n$ copies of $a$.

\item (Ivanov, p.\ 39.)  Prove that the symmetry group of an
equilateral triangle is isomorphic to the abstract group with two
generators $a$ and $b$ of order $2$ satisfying the additional
relation $aba=bab$.

\emph{Recall:} A group element $g$ is of order $n$ if $n$ is the
smallest natural number such that $g^n=e$.


% \item Let $G$ be a group, and let $H$ and $K$ be subgroups of $G$.
% Show that $H \cap K$ is also a subgroup of $G$.


\end{enumerate}
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