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\title{General Mathematics and CPS II}
\author{Exercise 8}
\date{February 27, 2015}
\maketitle

\begin{enumerate}

\item A \emph{Euclidean transformation} $F$ or, in short, a
\emph{motion} of the plane is a map which preserves the Euclidean
distance between points, i.e.
\[
  \lvert F(a) - F(b) \rvert = \lvert a - b \rvert
\]
where $\lvert a-b \rvert$ denotes the distance between two arbitrary
points $a$ and $b$.

Show that every motion of the plane is bijective.

\emph{Hint:} To prove that it is surjective, notice that the
transformation must map circles onto circles.

\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 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 $a^n=e$.  

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

\end{enumerate}
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