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\section*{Math 403/503 Spring 2024}
\subsection*{Homework 3, due February 7}
\emph{Updated on Feb 4, 4:30pm. The assumption that $N$ is normal was missing from Problem 1.}\\
\emph{Updated on Feb 7, 10:30am. In Problem 5, in the definition of $N_G(H)$, it should say $gHg^{-1}\boldsymbol{=}H$.}
\begin{enumerate}
\item The \emph{commutator subgroup} of a group $G$, denoted $[G,G]$ is the intersection of all normal subgroups of $G$ which contain $xyx^{-1}y^{-1}$ for all $x,y\in G$. Let $G$ be a group and $N$ be a \textbf{normal} subgroup. Show that $G/N$ is abelian if and only if $[G,G]\subseteq N$.
\smallskip
\item What is the size of the conjugacy class $\mathrm{Cl}_{S_8}(\sigma)$ if $\sigma=(123)(456)(78)$?
\smallskip
\item Let $G$ be a group. Let $\mathrm{Aut}(G)$ be the set of all automorphisms $\psi:G\to G$.
\begin{enumerate}[{\rm (a)}]
\item Show that $\mathrm{Aut}(G)$ is a subgroup of the symmetric group $S_G$.
\item For each $g\in G$, let $\phi_g:G\to G$ be the conjugation map $\phi_g(x)=gxg^{-1}$. Show that $\phi_g\in\mathrm{Aut}(G)$ for each $g\in G$.
\item Show that the function $\phi:G\to\mathrm{Aut}(G)$ given by $\phi(g)=\phi_g$ is a group homomorphism with kernel equal to the \emph{center} of $G$, defined as $Z(G)=\{z\in G\mid zg=gz\,\forall g\in G\}$.
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\smallskip
\item Let $G$ be a finite group. Suppose $H\leq G$, $N\trianglelefteq G$ and suppose $|H|$ and $|N|$ are coprime (i.e. have no common prime factors). Prove that $HN/N\cong H$. (Hint: Use Lagrange's Theorem and 2nd Isomorphism Theorem.)
\smallskip
\item Let $G$ be a group and $H\le G$. The \emph{normalizer of $H$ in $G$} is defined as $N_G(H)=\{g\in G\mid gHg^{-1}\boldsymbol{=} H\}$.
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\item Show that $N_G(H)\le G$ and $H\trianglelefteq N_G(H)$.
\item Show that if $B$ is any subgroup of $G$ such that $H\trianglelefteq B$, then $B\le N_G(H)$. (In other words, the normalizer is the largest subgroup of $G$ containing $H$ as a normal subgroup. Hence the name.)
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