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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 8.1  \\ Spring 2023 \hfill Due March 8}}}

\large

\vspace{1cm}

\begin{exercise}
Let $G$ be a group and let $H<G$.  Given $a \in G$, define
\[
\Lambda_a : G/H \to G/H
\]
by $\Lambda_a(xH) = (ax)H$.

\begin{itemize}

\item[\bf a.]  Show that $\Lambda_a$ is well-defined.  That is, if $x , y \in G$ and $xH=yH$, show that $(ax)H = (ay)H$.

\item[\bf b.]  Prove that $\Lambda_a \in \operatorname{Perm}(G/H)$. 

\item[\bf c.]  Prove that the map $\Lambda : G \to \operatorname{Perm}(G/H)$, defined by $a \mapsto \Lambda_a$ for $a \in G$, is a homomorphism.

\end{itemize}

\end{exercise}

\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 $f:G \to H$ be a homomorphism of groups and set $K = \ker f$.  Let $\overline{f}:G/K \to H$ be the induced map given by $\overline{f}(aK) = f(a)$, and let $\pi:G \to G/K$ be the canonical epimorphism.  Suppose that $\varphi : G/K \to H$ is a homomorphism satisfying the conclusion of the First Isomorphism Theorem, namely
\[
f = \varphi \circ \pi.
\]
Show that $\varphi = \overline{f}$.  This proves that $\overline{f} : G/K \to H$ is the unique homomorphism so that the diagram
\[
\xymatrix{
G \ar[r]^f \ar[d]_{\pi} & H \\
G/K \ar[ur]_{\overline{f}}
}
\]
commutes.  [{\em Suggestion.}  This is nearly trivial.  Given $aK \in G/K$, use the fact that $aK = \pi(a)$ to show that $\varphi(aK) = \overline{f}(aK)$.]
\end{exercise}

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