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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 9.1  \\ Spring 2019 \hfill Due March 27}}}

\large

\vspace{1cm}

\begin{exercise}
Let $G$ be a group and $H < G$.  Recall that for each $x \in G$, the left-translation map $T_x : G/H \to G/H$\footnote{Here $G/H$ simply indicates the set of left cosets of $H$ in $G$.  We are {\em not} assuming $H \lhd G$.} given by $yH \mapsto xyH$ is an element of $\operatorname{Perm}(G/H)$, and that the map $x \mapsto T_x$ is a homomorphism $G \to \operatorname{Perm}(G/H)$.   Let $K$ denote the kernel of this homomorphism.  

Assume that $G$ is finite and $[G:H] = p$ is the smallest prime dividing $|G|$.

\begin{itemize}

\item[\bf a.]  Use the first the First Isomorphism Theorem to show that $[G:K]$ divides $p$.

\item[\bf b.]  Use the fact that $K < H$ to show that $[G:K]$ is divisible by $p$.  

\item[\bf c.]  Conclude that $[G:K] = p$ and use this to show $H \lhd G$.

\end{itemize}

\end{exercise}

\begin{exercise}
Let $G$ be a nonabelian group of order 8.

\begin{itemize}

\item[\bf a.]  Explain why $G$ must have an element of order 4.

\item[\bf b.]  Let $x \in G$ have order 4 and $H = \langle x \rangle$.  Show that $[G:H] = 2$ and conclude that $H \lhd G$.

\item[\bf c.]  Let $y \in G \setminus H$.  Show that $G = \langle x , y \rangle$ and $y^2 \in H$.  Conclude that $y^2 = e$ and $|y| = 2$, or $y^2=x^2$ and $|y| = 4$.  [{\em Suggestion:}  Show that $y^2 = x$ and $y^2=x^3$ both imply $x^2 = e$.]

\item[\bf d.]  Prove that $yxy^{-1} = x$ or $yxy^{-1} = x^3$.  Show that the former is impossible since it implies $G$ is abelian. [{\em Suggestion:}  $H$ is normal in $G$ and conjugation preserves orders of elements.]  

\end{itemize}

\end{exercise}

\begin{exercise}
Let everything be as in the preceding exercise.

\begin{itemize}

\item[\bf a.]  If $y^2 = e$, prove that $G$ is isomorphic to the dihedral group
\[
D_4 = \langle r, f \, : \, |r| = 4, |f| = 2, frf = r^{-1} \rangle.
\]

\item[\bf b.]  If $y^2 = x^2$, prove that $G$ isomorphic to the quaternion group
\[
Q = \langle i, j \, : \, |i| = |j| = |ij| = 4, i^2 = j^2 = (ij)^2 \rangle.
\]

\end{itemize}

\end{exercise}


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