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\title{Group Theory: Math30038, Sheet 5}
\date{GCS}
\author{}
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\maketitle

\begin{enumerate}
\item Let $G = S_n$ be the symmetric group on $\{1,2,\ldots, n\}$, so
$|G| = n!$ and the elements of $G$ are the permutations of 
$\{1,2,\ldots, n\}$. \begin{enumerate}
\item Suppose that $(a_1, a_2, \ldots , a_t) \in G$ is a cycle, and
that $g \in G$. Show that $g^{-1}(a_1, a_2, \ldots , a_t)g
= (a_1g, a_2g, \ldots , a_tg).$
\item Each element of $G$ can be expressed as a product of disjoint cycles
(elements of $G$ are {\em disjoint} if their supports are disjoint). Show 
that the number of conjugacy classes in $S_n$ is the number of
ways of writing $n$ as an ascending sum $a_1 + a_2 + \cdots + a_t$ of positive 
integers $a_1 \leq a_2 \leq \cdots \leq a_t$. {\em Thus there are 3 
conjugacy classes in $S_3$ because 3 can be written as an ascending sum
in three ways: $3, 1+2, 1+ 1 +1.$ Also in $S_4$ there are 5 conjugacy classes because
4 is $4, 1+3, 1+1 +2, 2+2$ and $1 + 1+ 1 + 1$.}
\item Determine the number of conjugacy classes in $S_5$, and the size of each
conjugacy class, and describe the centralizer in $G$ of a chosen representative
of each conjugacy class.
\end{enumerate}
\item Let $G = S_n$. Let $x = (1,2,\ldots, n) \in G$. Prove that
$C_G(x) = \langle x \rangle.$
\item Show that in $S_4$ there is a non-identity element $y$ such that
$C_G(y) \not = \langle y \rangle.$
\item Suppose that $G$ is a finite group. Show that the number of elements
in each conjugacy class of $G$ must divide $G$.
\item Let $G$ be a group with a subgroup $H$ such that
$g^{-1}H g \subseteq H$ for every $g \in G$. Prove that
$g^{-1}H g = H$ for every $g \in G$.
\item (Challenge) Does there exist a group $G$ containing an element $g$
and a subgroup $H$ such that $g^{-1}Hg \subseteq H$ but $g^{-1}H g \not
= H$.
\item Suppose that $G$ is a group and that $H \leq G$. Choose $g \in G$.
Prove that $g^{-1}Hg \leq G$.
\item Let $G$ be a finite group of order $n$ which has $t$ conjugacy classes.
Elements $x$ and $y$ are each selected uniformly at random from $G$. What is
the probability that $x$ and $y$ commute? Does this make sense for abelian groups?
\item Show that a finite group with exactly two conjugacy classes must have two elements.
\item Let $G$ be a containing $H$ a subgroup of finite index. Let
$S = \{x^{-1}Hx \mid x \in G\}.$ Let $G$ act on $S$ by conjugation so
if $K \in S$ then $K \cdot g = g^{-1}Kg$. Verify that this is a group action,
and deduce that $|S| = |G:N_G(H)|$ where 
$N_G(H) = \{ g \in G \mid gH = Hg \}.$ Deduce that $|S|$ is finite and 
divides $|G:H|$.
\end{enumerate}
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