\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}[Left group action]
Given a group $(G,\circ)$, 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 $g \cdot s$.

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.

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


\begin{proposition}
$\alpha_g (s)$ is bijective.
\end{proposition}


We provide some exmaples of group actions.

\begin{example}[Left regular action]
\[ \alpha : G \times G \to G \]
\[ \alpha (g,h)= gh \]
\end{example}


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


Noting that the actions 'group' here is $H$ and 'set' is $G$, we can think of left cosets as orbits of the following left action.
\begin{example}[Left coset projection action]
\[ \alpha : H \times G \to G \]
\[ \alpha (h,g)= gh^{-1}\]
\end{example}



Note that we invert $h$ so that $\alpha (h_1, \alpha (h_2,g) ) = \alpha (h_1 h_2 , g)$ is satisfied (since $h_{2}^{-1}h_{1}^{-1} = (h_{1} h_{2})^{-1}$.



\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}





\begin{example}
\[ \alpha : G \times G  / H \to G/ H \]
\[ \alpha (g_1,gH)= (g_1 g)H \]
\end{example}




\section{Orbits}

We know that if we fix the first (group) argument of an action, the image is the set $S$ it works over (since such functions are bijections), but what if we fix the second (set) argument and then consider its image? This is a more interesting situation, and the image of such a function is the idea of an \emph{orbit}.

Simply put, orbits represent all possible actions on a set element. They are a generalization of the notion of a left coset.

\begin{definition}[Orbit of a set element]
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 $s$.
\[\mathrm{Orb}_{\alpha}(s) = G \cdot s = \{ \alpha(g, s) : g \in G \}\]
The set of unique orbits on each set element is denoted $S / G$.
\[ S / G = \{ G \cdot s : s \in S\} \]
\end{definition}


We have already seen examples of these hidden; in the left coset projection action, the orbits are the left cosets formed by each $g$. 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 $s$.




Let's now discuss the elementary properties of orbits, we will see some similar properties to left cosets. Indeed, some of the properties we had discovered about left cosets were really just general properties group actions!
Similarly to the idea of how cosets partition a group, orbits partition a generic set.

\begin{proposition}
\[ y \in \mathrm{Orb}_{\alpha} (x) \iff  \mathrm{Orb}_{\alpha} (x) = \mathrm{Orb}_{\alpha} (y) \]
\end{proposition}
Having $y \in \mathrm{Orb}_{\alpha} (x)$ means some $g$ satisfies $g \cdot x = y$  and so $x = g^{-1} \cdot y$, so we also have $x \in \mathrm{Orb}_{\alpha} (y)$ and hence $\mathrm{Orb}_{\alpha} (x) = \mathrm{Orb}_{\alpha} (y)$.
The converse is easily true because $y \in \mathrm{Orb}_{\alpha} (y)$, but since  $\mathrm{Orb}_{\alpha} (y) = \mathrm{Orb}_{\alpha} (x)$, we naturally have $y \in \mathrm{Orb}_{\alpha} (x)$ since they are the same set.

\begin{proposition}
Orbits are either equal or disjoint.
\[G \cdot s \neq G \cdot t \implies G \cdot s \cap G \cdot t =\emptyset\]
\[ \mathrm{Orb}_{\alpha} (x) \neq \mathrm{Orb}_{\alpha} (y) \implies \mathrm{Orb}_{\alpha} (x) \cap \mathrm{Orb}_{\alpha} (y) = \emptyset\]
\end{proposition}

Having $\mathrm{Orb}_{\alpha} (x) \neq \mathrm{Orb}_{\alpha} (y)$, we have $y \notin \mathrm{Orb}_{\alpha} (x)$. Since for any arbitrary $y_i \in \mathrm{Orb}_{\alpha} (y)$ we have $\mathrm{Orb}_{\alpha} (x) \neq \mathrm{Orb}_{\alpha} (y_i)$ (since we know that $y$ and $y_i$ generate the same orbit) and hence $y_i \notin \mathrm{Orb}_{\alpha} (x)$. Since $y_i$ was an arbitrary element, the orbits are disjoint.

Now assume $y \notin \mathrm{Orb}_{\alpha} (x)$, we show that a contradiction occurs if $\mathrm{Orb}_{\alpha} (x) \cap \mathrm{Orb}_{\alpha} (y)$ is non-empty. 


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


\begin{proposition}
\[S = \bigcap_{s \in S} G \cdot s\]
\end{proposition}


Since the distinct cosets partition a group, we obtained Lagrange's theorem.  Since distinct orbits partition a set, a weaker but more general result holds for group actions. Since orbits do not generally exhibit the same bijective behaviour as cosets, different orbits may have different cardinalities, so the cardinality of an orbit may doesn't divide the cardinality of the set like cosets do, but at least we know that the distinct orbits sum up to the set.

The following result is a generalization of Lagrange's theorem of cosets to a group from actions to a set. Indeed, applying the result on the left-coset action $\alpha(h,g)=gh^{-1}$ and using the fact that for any $g_1,g_2 \in G$ we have $|g_1 H|=|g_2 H|$ is exactly Lagrange's theorem!

\begin{corollary}
Consider the following equivalence classes that capture elements with the same orbit.
\[ [x] = \{ y \in S : Gy=Gx \} \]
Then the following $E$ is used to index the distinct orbits.
\[ B = \{ [x] : x \in S \}\]
Then the following holds.
\[|S| = \sum_{ [x] \in B} |Gx|\]
\end{corollary}

Later on we will find another formulation of this formula in terms of another object called a \emph{stabilizer} which we will see shortly.

\section{Stabilizers}


When considering actions, it is of interest to examine the fixed points of the action (i.e the input set element is the output set element).

\begin{definition}[Fixed point of $G$]
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 $S^G$.
	\[ \mathrm{Fix}_{\alpha}(G) = \{ s \in S : \forall g \in G [ \alpha(g,s)=s ] \}\]
We can also consider the fixed points with respect to a group subset.
	\[ \mathrm{Fix}_{\alpha}(H) = \{ s \in S : \forall h \in H \leq G [ \alpha(h,s)=s ] \}\]
\end{definition}

This set gives us the fixed points of the group action, however what if we want to see group elements can be fixed to make some set element $x$ (or better yet, any element in some $X$) act as fixed points?

In a similar vein to orbits, we can fix the second (set) argument and see what group elements produce fixed points for that element, this is the notion of a \emph{stabilizer}.

Simply put, stabilizers represent all group elements for which a set element is a fixed point (or for which a collection of set elements are all fixed points). 

\begin{definition}[Stabilizer of $X$]
Given a set element $x$, the \emph{stabilizer of $x$} is the set of all group elements for which $x$ is a fixed point.
\[ \mathrm{Stab}_{\alpha}(x) = G_x = \{ g \in G : \alpha (g,x) = x\}\]
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 $X$.
\[\mathrm{Stab}_{\alpha}(X)  =  G_X = \{ g \in G : \forall x \in X [\alpha (g,x) \in X ]\}\]
\end{definition}



\begin{proposition}
	\[ \mathrm{Stab}_{\alpha}(x)  \leq G \]
\end{proposition}

Unfortunately even though it forms a subgroup, it may not always form a normal subgroup, meaning we can't always make quotient groups from them. For the sake of curiosity, I have fished out a necessary and sufficient condition when they are normal subgroups.

\begin{proposition}
Let $G$ be a group. A stabilizer $\mathrm{Stab}_{\alpha}(x)$ is a normal subgroup iff every set element in the orbit $\mathrm{Orb}_{\alpha}(x)$ forms the same stabilizer as $\mathrm{Stab}_{\alpha}(x)$.
\[ \mathrm{Stab}_{\alpha}(x)  \lhd G \iff \forall \omega \in \mathrm{Orb}_{\alpha}(x) [ \mathrm{Stab}_{\alpha}(x)=\mathrm{Stab}_{\alpha}(\omega) ] \]
\end{proposition}

Even though the set of left cosets of stabilizers doesn't always form a quotient group, we will study it anyways because it has an interesting connection to orbits by the orbit stabilizer theorem!



\section{Orbit-Stabilizer theorem}


Orbits and stabilizers have an interesting combinatorial relation between them. We start by comparing the behaviour of $gG_{x}$ and $g \cdot x$ with different $g$ and notice the following insight; when $gG_x=hG_x$ we also have $g \cdot x = h \cdot x$.

As it turns out, there is a bijection between the set of $x$-stabilizer left cosets and $x$-orbits!


\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}

We obtain a juicy corollary from the orbit-stabilizer theorem since a bijective function between sets implies equal cardinalities of domain and codomain.

By taking the cardinality of the domain and image of this bijection and applyting Lagrange's theorem, we get an interesting result.
\begin{corollary}
\[ |G / \mathrm{Stab}_{\alpha}(x)| = |\mathrm{Orb}_{\alpha}(x)| \]
Futhermore, we can apply Lagrange's theorem to derive the following.
\[ |G| = |\mathrm{Orb}_{\alpha}(x)||\mathrm{Stab}_{\alpha}(x)| \]
\end{corollary}

The cardinality of an $x$-orbit is the amount of distinct cosets of the $x$-stabilizer. Not only that, but the use of Lagrange's theorem tells us that the cardinality of an $x$-orbit divides the order of the group! $x$-stabilizers are subgroups, hence Lagrange implies this automatically, however $x$-orbits require this orbit-stabilizer theorem for such a result.


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

We expressed our  'generalized Lagrange's theorem' in terms of orbits, but due to the orbit-stabilizer theorem and its corollary we can translate our statement to be in terms of stabilizers instead.

\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.

Consider the following equivalence classes that capture elements with the same orbit.
	\[ [x] = \{ y \in S : G / \mathrm{Stab}_{\alpha}(y) = G / \mathrm{Stab}_{\alpha}(x) \} \]
Then the following $E$ is used to index the distinct orbits.
\[ B = \{ [x] : x \in S \} \setminus \{ [x] : x \in \mathrm{Fix}_{\alpha}(G) \} \]
Then the following holds.
	\[ |S| = |\mathrm{Fix}_{\alpha}(G)| + \sum_{[x] \in B} |G / \mathrm{Stab}_{\alpha}(x)| \]
\end{corollary}



\subsection{Burnside's lemma}

Group actions are very powerful for creating combinatorial arguments due to its ability to generalize Lagrange's theorem from a group to any set that is compatible with a desired group action. Extending on those results leads us to a prominent result known as \emph{Burnside's lemma}.



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]
\[ |S / G|  = \frac{\sum_{g \in G}|\mathrm{Fix}_{\alpha}(\{g\})|}{|G|}\]
\end{lemma}



\section{Conjugacy}

We have sufficient background in group actions to look at applications to the study of groups. Although we have studied left cosets without resorting to group actions, group actions prove extremely useful in studying the congugacy action, which like the left coset projection action, can be defined naturally on any group.

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.

\begin{definition}[Conjugacy class of $h$]
The \emph{conjugacy class of $h$} is the set of elements possible by conjugating $h$ by group elements.
\[ C(G,h) = \{ ghg^{-1} \in G : g in G \} \]
%Conjugacy classes are the orbits of the conjugation action.
\end{definition}

The conjugacy class of $\{h\}$ is the orbit $G \cdot h$ by the conjugacy action.


\begin{definition}[Centralizer of a group subset]
The \emph{centralizer of $S$ in $G$} is the set of elements that commute with the elements of $S$.
%Since $gh=hg \implies ghg^{-1}=h$, it also happens to be the stabilizer for conjugation.
%\[ C(G,s) = \{ g \in G :  gs=sg ]  \} \]
\[ Z(G,S) = \{ g \in G : \forall s \in S \subseteq G [ gsg^{-1}=s ]  \} \]
%Centralizers are the stabilizers of the conjugation action.
\end{definition}

The centralizer of $\{h\}$ is the stabilizer $\mathrm{Stab}_{\alpha}(\{h\})$ for the conjugacy action.

Note also that the set of all fixed points (with respect to conjugacy action) is therefore just the center $Z(G)$
\[Z(G,G)=Z(G)=\mathrm{Fix}_{\alpha}(G)\]



\begin{definition}[Normalizer of a group subset]
The \emph{normalizer of $S$ in $G$} is the set of elements that make $S$ conjugate to itself.
\[ N(G,S) = \{ g \in G :  gSg^{-1}=S  \} \]
%normalizers are the stabilizers of subgroups with respect to the conjugation action.
\end{definition}


The normalizer of $S$ is the stabilizer $\mathrm{Stab}_{\alpha}(S)$ for the conjugacy action.



Consider the normalizers of a singleton $\{h\}$ (so $N(G,\{h\}) = \{g \in G : ghg^{-1}=h\}$), since $gh=hg \iff ghg^{-1}=h$, such normalizers happens to be the stabilizer for the conjugation action.





By applying one of our counting theorem with the conjugacy action, we get the following formula to enumerate any group.


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







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

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

By considering the generalized Lagrange theorem on $p$-groups, we notice the following proposition and corollary.
\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}



\section{Sylow theorems}


\begin{definition}[Sylow $p$-subgroup]
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}.
\[p \text{ is prime }, \mathrm{gcd}(p,m)=1, n \in \mathbb{N} \setminus \{0\}  \]
\[H \text{ is a Sylow p-subgroup of } G \iff H \leq G \land |G|=p^n m \land |H|=p^n\]
\end{definition}


\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]
\[ |\mathrm{Syl}_p(G)| \equiv 1 \mod p \]
\[ m \equiv 0 \mod |\mathrm{Syl}_p(G)| \]
	\[ |\mathrm{Syl}_p(G)| = | G / N(P) |, P \in \mathrm{Syl}_p (G) \]
\end{theorem}


-Sylow p-group


Recall our beloved Lagrange's theorem; it's pretty cool that the order of subgroups divide the order of the group. However, if a number divides the order of a group, does that mean a subgroup with an order of that number be made? This is the converse to Lagrange's theorem, and it is not true in general, however it turns out that this is true for prime numbers and these groups happen to be cyclic!

\begin{proposition}[Cauchy's theorem]
Let $G$ be a finite group. If a prime number $p$ divides the order of $G$, then there exists a cyclic subgroup of $G$ with order $p$.
\end{proposition}



\section{Wreath product}