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\title{Math 120 WIM project}
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The \textit{Orbit-Stabilizer Theorem} is an important fact
that underlies much of group theory. A version appears in
Proposition~2 on page 114 of Dummit and Foote.

Please read \textit{Writing Mathematics} posted on the course web page, which
describes a style of writing mathematics that I would like you to try to use.
In this document, I will state three results that I would like you to write up.
You will be graded on good writing style as well as the correctness of the
math. Do not use ChatGPT or other AI. You may \textit{discuss} the
project with other students or instructors but do not show other students your
written work. You are strongly recommended to use \LaTeX.

You may include some examples or other expository material explaining why the
following results are important, and give proofs for the following 3 results.
Define the terms and notations that you use. In particular, define the terms
\textit{orbit}, \textit{stabilizer} and \textit{transitive}. Also define the
following term: if $G$ is a group acting on sets $X$ and $Y$ then the actions
are \textit{equivalent} if there is a bijection $\phi:X\to Y$ such that for
$x\in X$ and $g\in G$ we have $\phi(g\cdot x)=g\cdot\phi(x)$.

A particular action: let $H$ be a subgroup of $G$ (not necessarily
normal) and let $G/H$ denote the set of left cosets $x H$. Then
$G$ acts on $G/H$ as follows: $g\cdot x H= gx H$.

\begin{theorem}[The \textit{Orbit-Stabilizer Theorem}]
\label{thm:ost}
Let $G$ be a finite group acting transitively on a set $X$. Let $a\in X$ and
let $H$ be the stabilizer of $a$. Then the action on $X$ is equivalent to the
action of $G$ on $G/H$.  
\end{theorem}

Now let $G$ be a finite group acting on itself by
conjugation. Using Theorem~\ref{thm:ost}, prove:

\begin{proposition}
\label{prop:xconj}
If $G$ is a finite group and $x\in G$ be an element of order $k$.
Then $k$ divides the order $|G|$ and the number of conjugates of 
$x$ in $G$ divides $|G|/k$.
\end{proposition}

Let $p$ be a prime. A group $G$ is called a \textit{$p$-group} if $|G|$
is a power of $p$. Use Proposition~\ref{prop:xconj} to prove:

\begin{proposition} Assume that $|G|=p^k$ for $k>0$ and
let $Z$ be the center of $G$. Then $|Z|$ is a multiple of $p$.
\end{proposition}

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