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\title{General Mathematics and ACM II}
\author{Exercise 7}
\date{February 25, 2011}
\maketitle

\begin{enumerate}

\item Use the matrix form of the equation for a reflection (see
handout) to show that the composition of reflections about parallel
lines is a translation $\Pi_{\boldsymbol v}$.  Find an expression for the
translation vector $\boldsymbol v$.

\item Let $G$ be a group and let $a,b \in G$.  Show that
$(ab)^{-1}=b^{-1} \, a^{-1}$.

\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 $a^n=e$.  (Where $a^n$ is understood as letting
the group operation act between $n$ copies of $a$.)


\item \emph{(If not submitted on Friday.)}

(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. 

\end{enumerate}
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