\part{Group Actions}

\chapter{Group Actions}

\section{Group Actions}

Groups have thus far been very interesting to study in and of themselves, however groups are equally interesting to study on how they can interact with generic sets. The groups $\mathrm{Sym}(n)$ have permutation functions as their object, however these permutations are functions operating on $\mathbb{N}\cap[1,n]$. 

This is quite a motivating example to develop some theory of how group elements can 'act' on some set; imagine we want to prove things about permutations that don't move $5$ around? Some of this we've been doing implicitly (mutually exclusive permutations), however this idea can be generalized. 

\begin{definition}
	Given a group $(G,\cdot)$, a \emph{left group action} $\alpha : G \times S \to S$ is a map $\alpha (g,s) = g \cdot s$ with the following properties for any $s \in S$ and $g,h \in G$.
\begin{itemize}
	\item $\alpha ( 1_G , s) = s$
	\item $\alpha ( g , \alpha (h, s)) = \alpha (gh,s)$
\end{itemize}
We say that group $G$ acts on $S$.
\end{definition}

When the group action and set element is apparent, a group action $\alpha (g,s)$ may be denoted as $g \cdot s$ or $gs$. It is important to note that in this context we read $\cdot$ as the group action; not the group operation!


It is sometimes useful to 'curry' group actions; this means that instead of interpreting it as a group element and set element to make a set element, we can think of group actions as group elements making functions from the set to itself. These functions can be proven to be bijective.

Instead of one bivariate function, we have a class at most $|G|$ univariate functions.

\[\alpha_g :  S \to S\]
\[ \alpha_g (s) = \alpha(g,s)\]


We provide some exmaples of group actions.

\begin{example}[Group operation from the left]
\[ \alpha : G \times G \to G \]
\[ \alpha (g,h)= gh \]
\end{example}


\begin{example}[Conjugation]
\[ \alpha : G \times G \to G \]
\[ \alpha (g,h)= ghg^{-1} \]
\end{example}


\begin{example}[Cosets]
\[ \alpha : H \times G \to G \]
\[ \alpha (h,g)= gh^-1\]
\end{example}


\begin{example}[Permutation functions]
\[ \alpha : \mathrm{Sym}(n) \times \mathbb{N}\cap[1,n]  \to \mathbb{N}\cap[1,n] \]
\[ \alpha (\sigma,i)= \sigma(i) \]
\end{example}



\section{Orbits}

Much like the idea of how cosets are a way to study behaviour of some $g$ on a subgroup $H$, we can study the behaviour of a group action with various group arguments on a set element $s$. These are called \emph{orbits}.

\begin{definition}
Given a group action $\alpha : G \times S \to S$, the \emph{orbit of $s$} is the set of element obtainable by $G$ acting on a specific $s$.
\[ Gs = \{ \alpha(g, s) : g \in G \}\]
The set of unique orbits on each set element is denoted $S / G$.
\[ S / G = \{ Gs : s \in S\} \]
\end{definition}

Similarly to the idea of how cosets partition a group, orbits partition a generic set.

\begin{proposition}
Orbits are either equal or disjoint.
\[S = \bigcap_{s \in S} Gs\]
\[Gs \neq Gt \implies Gs \cap Gt =\emptyset\]
\end{proposition}

\begin{corollary}Orbits are equivalence classes on $X$.
\end{corollary}



\begin{definition}
A \emph{fixed point of $G$} is some element $s$ such that for any $g\in G$,$g \cdot s = s$.
The set of fixed points of $S$ under $G$ is denoted $S^G$.
\[ S^{G} = \{ s \in S : \forall g \in G [ \alpha(g,s)=s ] \}\]
\end{definition}



\begin{definition}
Given a set element $x$, the \emph{stabilizer of $x$} is the set of all group elements for which $x$ is a fixed point.
\[G_x = \{ g \in G : \alpha (g,x) = x\}\]
Given a subset $X \subseteq S$, the \emph{stabilizer of $X$} is the set of all group elements for which every action on an $x \in X$ is stil in $X$.
\[G_X = \{ g \in G : \forall x \in X [\alpha (g,x) \in X ]\}\]
\end{definition}

For this section, we will only be concerned with stabilizers of elements.

Similar to the kernel of a homomorphism, stabilizers form a subgroup.

\begin{proposition}
\[ G_x  \leq G \]
\end{proposition}


The next proposition informally means that the larger the stabilizer is, the smaller the orbits are. It also means there are just as many unique cosets of the stabilizer of $x$ as elements in the orbit of $x$.

\begin{theorem}[Orbit-Stabilizer theorem]
Let $f : G / G_x \to Gx$ be a function defined by $f(gG_{x_0}) = \alpha(g,x_0)$. $f$ is bijective; $x$ stabilizer cosets are in bijection with elements of the orbit of $x$.
\end{theorem}

When I was learning group theory, I stumbled across the following result while playing around

\begin{proposition}
Let $G$ be a group. A stabilizer $G_x$ is a normal subgroup iff every set element in the orbit $Gx$ forms the same stabilizer as $G_x$.
	\[ G_x  \lhd G \iff \forall \omega \in Gx [ G_x=G_\omega ] \]
\end{proposition}

\begin{example}
	Given $H \leq G$, let $\alpha : H \times G \to G$ be the group action defined by $\alpha(h,g) = g \cdot h^{-1}$. The orbits of this group action are simply the left cosets. The mapping $g\cdot h$ exhibits the same behaviour but conflicts with the required property of group actions that $\alpha(h_2,\alpha(h_1,g)) = \alpha(h_2 \cdot h_1, g)$. All the stabilizers are $G_x = \{1_G\}$ for this action.
\end{example}


\section{Burnside's lemma}

Group actions are very powerful for creating combinatorial arguments, among the most prominent being \emph{Burnside's lemma}.

We have established that orbits are either equal or disjoint, and elements of orbits  of $x$ have a bijection to cosets of the stabilizer of $x$. means that the sum of these stabilizer coset cardinalities is the sum of the orbit cardinalitites.

\begin{corollary}
Let $G$ be a finite group acting on $S$, then the following holds where $B$ is a set constructed by choosing one element from each orbit with over one element.
\[ |S| = |S^G| + \sum_{x \in B} |G / G_x| \]
\end{corollary}


Using this corollary, one can derive \emph{Burnside's lemma}. It states that we can count the amount of unique orbits by finding the 'average' of each '$g$-invariant set of elements' (set elements where the group action with $g$ has no effect). It is an indispensable tool in combinatorics; particularly for counting how many symmetrically unique colorings there are on an $n$-gon.

\begin{lemma}[Burnside's lemma]
	\[ |S / G|  = \frac{\sum_{g \in G}|S^g|}{|G|}\]
	\[ S^g = \{s \in S : gs=s\}\]
\end{lemma}



\section{Conjugacy classes}

Let's now Consider group actions onto groups.
Recall the action $\alpha(g,h)=ghg^{-1}$ is called \emph{conjugation}. The \emph{conjugacy class of $h$} is the orbit of $h$ of conjugation.
\[ C(h) = Gh = \{ ghg^{-1} : g in G \} \]

\begin{definition}
The \emph{centralizer of $h$} is the set of elements that commute with $h$. Since $gh=hg \implies ghg^{-1}=h$, it also happens to be the stabilizer for conjugation.
\[ Z(h) = G_h = \{ g \in G : gh=hg \} \]
\end{definition}

Note also that the set of all fixed points $G^G$ is therefore just the center $Z(G)$


\begin{proposition}
Let $G$ be a finite group, then the following holds where $B$ is a set constructed by choosing one element from each conjugacy class with over one element.
	\[  |G| = |Z(G)| +\sum_{h in B} |G /Z(h)|\]
\end{proposition}

Let's take a look at the stabilizers of subgroups; stabilizers under the conjugation action create normal subgroups. Hence, they are called \emph{normalizers}.
\begin{definition}
If one considers sugbroups $H \leq G$, the set of $G_H$ \emph{normalizer of $H$}.
\[N_G (H) = G_H = \{g \in G : gHg^{-1}=H \}\]
\end{definition}



\begin{corollary}
	\[ |G| = |Z(G)| + \sum_{h : C(h) \in G / G \land |C(h)| >1} |G / Z(h)| \]
\end{corollary}


\chapter{Groups of order $p^r$}

\begin{definition}
A \emph{$p$-group} is a group of order $p^n$.
\end{definition}

\begin{proposition}
Let $G$ be a $p$-group with order larger than 1 acting on a finite set $S$, then the following holds.
\[ |S| \equiv |S^G| \mod p \]
\end{proposition}


\begin{corollary}
Let $G$ be a finite $p$-group with order larger than 1, then the following holds.
\[ |G| \equiv |Z(G)| \mod p \]
\end{corollary}

\begin{corollary}
Let $G$ be a finite $p$-group with order $p^2$, then $G$ is Abelian.
\end{corollary}


\chapter{Sylow theorems}


\begin{definition}
Let $G$ be a finite group of order $p^n m$ where $p$ is prime and $m$ is coprime to $p$, a subgroup of $G$ with order $p^n$ is called a \emph{Sylow $p$-subgroup}.
\end{definiton}


\begin{theorem}[First Sylow theorem]
Let $G$ be a finite group, for each prime factor $p$ of $|G|$, $G$ has a Sylow $p$-subgroup.
\end{theorem}

\begin{theorem}[Second Sylow theorem]
Let $H,K$ be two Sylow $p$-subgroups of the finite group $G$, then $H$ and $K$ are conjugate to eachother. That is, there exists a $g\in G$ where the following holds.
\[ gHg^{-1}=K \]
\end{theorem}


\begin{theorem}[Third Sylow theorem]
\[ |\mathrm{Syl}_p(G)| \equiv 1 \mod p \]
\[ m \equiv 0 \mod |\mathrm{Syl}_p(G)| \]
\end{theorem}


-Sylow p-group
