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\title{General Mathematics and Computational Science II}
\author{Exercise 8}
\date{February 27, 2007}
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\begin{enumerate}

\item (Ivanov, p.\ 39.)  Recall that the symmetry group of a subset
$A$ of the plane is defined as
\[
  \operatorname{Sym}(A) = \{ F \text{ motion} \colon F(A)=A \} \,.
\]
Prove that such a set of motions is indeed a group. 

\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 $g^n=e$.

\emph{Hint:} Count the number of elements of this group.

\end{enumerate}
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