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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 7.3  \\ Spring 2019 \hfill Due March 6}}}

\large

\vspace{1cm}

\begin{exercise}
Let $G$ be a group and $H<G$.  The {\em normalizer of $H$ in $G$} is
\[
N_G(H) = \{ x \in G \, | \, xHx^{-1} = H \}.
\]
\begin{itemize}

\item[\bf a.]  Prove that $N_G(H)$ is a subgroup of $G$ containing $H$, and that $H$ is normal in $N_G(H)$.

\item[\bf b.]  Prove that the set $\{xHx^{-1} \, | \, x \in G \}$ of conjugates of $H$ is in one to one correspondence with the left cosets of $N_G(H)$ in $G$.

\end{itemize}
\end{exercise}

\begin{exercise}
Let $G$ be a group and $H<G$.  The {\em centralizer of $H$ in $G$} is
\[
Z_G(H) = \{ x \in G \, | \, xy = yx \, \text{ for all } y \in H\}.
\]
Prove that $Z_G(H)$ is a normal subgroup of $N_G(H)$.
\end{exercise}

\begin{exercise}
Let $G$ be a group, $H < G$.  Recall the homomorphism $T: G \to \operatorname{Perm}(G/H)$ of Lang, II.4.28, which is defined by $x \mapsto T_x$, where $T_x(yH) = xyH$.  Prove that
\[
\ker T = \bigcap_{x \in G} xHx^{-1},
\]
the so-called {\em normal core} of $H$.
\end{exercise}

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