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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 1.1  \\ Spring 2023 \hfill Due January 18}}}

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\vspace{1cm}

\begin{exercise}
If $S$ is a set and $\circ$ denotes composition of functions, prove that $(\operatorname{Perm}(S),\circ)$ is a group.  Show that $\operatorname{Perm}(S)$ is abelian if and only if $|S| \le 2$.
\end{exercise}

\begin{exercise}
Let $G$ be a group.  Given $a \in G$, define $\lambda_a : G \to G$ by $\lambda_a(g) = ag$ for all $g \in G$.  The function $\lambda_a$ is sometimes called {\em (left) translation by $a$}.  

\begin{itemize}

\item[\bf a.] Prove that for any $a \in G$ one has $\lambda_a \in \operatorname{Perm}(G)$.  

\item[\bf b.] Prove that for all $a,b \in G$, $\lambda_a = \lambda_b$ if and only if $a=b$.  [{\em Suggestion:}  Evaluate both functions at the identity.]

\end{itemize}
\end{exercise}

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