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{\bf \Large 250A Homework 6}\\[7mm]
{\small Due Monday November 15}\\[4mm]
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\noindent
{\bf Question 1} 
Let $G$ act on the set $S$. Show that if $s$, $t$ $\in$ $S$ belong
to the same orbit, their stabilizers are conjugate subgroups
of $G$.

\vspace{.4cm}\noindent
{\bf Question 2} 
Let $x$ and $y$ be conjugate elements in a finite group
$G$. Show that the number of distinct $g\in G$ with $g^{-1}xg=y$
equals the order of the normalizer of $x$.


\vspace{.4cm}\noindent
{\bf Question 3}
Which finite groups have precisely one or two conjugacy classes?
 
\vspace{.4cm}\noindent
{\bf Question 4} 
Fix $x\in G$ a group. Show that $G\ni g\mapsto xgx^{-1}\in G$
is an automorphism of $G$ and in turn demonstrate that
$G/Z(G)$ is isomorphic to a subgroup of ${\rm Aut}(G)$. 
 
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