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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 10.2  \\ Spring 2023 \hfill Due March 29}}}

\large

\vspace{1cm}


\begin{exercise}
Let $G$ be a group.  Given $a,b \in G$ we say that $a$ is {\em conjugate} to $b$ provided there is an $x \in G$ so that $xax^{-1} = b$.  Prove that conjugacy (being conjugate to) defines an equivalence relation on $G$.  The equivalence classes under this relation are called the {\em conjugacy classes} of $G$.
\end{exercise}

\begin{exercise}
Prove that $H = \{\operatorname{id}, (12)(34), (13)(24), (14)(23) \}$ is a normal subgroup of $S_4$.  Since $H \le A_4$, this proves that $H \triangleleft A_4$, and hence that $A_4$ is {\em not} simple.  Prove that $S_4/H \cong S_3$ and $A_4/H \cong \Z_3$.
\end{exercise}

\begin{exercise}
Carefully prove that any two $r$-cycles are conjugate in $S_n$ (the proof I gave in class wasn't quite correct).
\end{exercise}

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