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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 4.3  \\ Spring 2019 \hfill Due February 13}}}

\large

\vspace{1cm}

\begin{exercise}
Let $f: G \to H$ be a homomorphism of groups.

\begin{itemize}

\item[\bf a.]  If $J < G$, prove that $f(J) < H$.

\item[\bf b.]  If $K < H$, prove that $f^{-1}(K) < G$.\footnote{Recall that given any function $f:X \to Y$ and any subset $S \subset Y$, the {\em inverse image} of $S$ under $f$ is \[f^{-1}(S) = \{ x \in X \, | \, f(x) \in S \} .\]  The inverse image makes sense whether or not $f$ is invertible.  However, when $f$ does happen to be invertible, $f^{-1}(S)$ is indeed the same as the forward image of $S$ under $f^{-1}$, as the notation suggests.}

\end{itemize}

\end{exercise}

\begin{exercise}
Let $G$ be a group and let $\operatorname{Aut}(G)$ denote the set of automorphisms of $G$.

\begin{itemize}

\item[\bf a.]  Prove that  $\operatorname{Aut}(G) < \operatorname{Perm}(G)$.

\item[\bf b.]  Given $x \in G$, define $c_x : G \to G$ by $c_x(y) = xyx^{-1}$.  Prove that $c_x \in \operatorname{Aut}(G)$.  The map $c_x$ is called {\em conjugation by $x$}.

\item[\bf c.]  Prove that the map $x \mapsto c_x$ from $G$ to $\operatorname{Aut}(G)$ is a homomorphism.  Its image is subgroup of {\em inner automorphisms} of $G$.  

\end{itemize}

\end{exercise}

\begin{exercise}
Let $\phi : \Q \to \Q$ be an endomorphism of $\Q$.  Prove that there exists an $a \in \Q$ so that $\phi(r) = ar$ for all $r \in \Q$.  [{\em Suggestion:}  Consider $\phi(1)$.]
\end{exercise}

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