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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 8.2  \\ Spring 2019 \hfill Due March 20}}}

\large

\vspace{1cm}

\begin{exercise}
Let $f:X \to Y$ be a function.

\begin{itemize}

\item[\bf a.]  Let $A \subset X$.  Prove that $A \subset f^{-1}(f(A))$, with equality if $f$ is injective.

\item[\bf b.]  Let $B \subset Y$.  Prove that $f(f^{-1}(B)) \subset B$, with equality if $f$ is surjective.

\end{itemize}

\end{exercise}

\begin{exercise}
Let $a$ and $b$ be elements of a group $G$.  If $a$ has odd order and $aba^{-1} = b^{-1}$, prove that $|b| = 2$.  [{\em Suggestion:}  Notice that the hypotheses imply that conjugation by $a$ is an automorphism of $\langle b \rangle$ of order 2.]
\end{exercise}

\begin{exercise}
Let $G$ be a finite abelian group, $n \in \N$, and suppose that $\gcd(|G|,n)=1$.  Prove that every element of $G$ has a unique $n$th root.  [{\em Suggestion:}  Reformulate the conclusion in terms of the map $x \mapsto x^n$.]
\end{exercise}

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