Definition 5.23. Orbit.
Let \(G\) be a group acting on a set \(X\text{.}\) For an element \(s\in X\) the orbit of \(s\) is
\begin{equation*}
\operatorname{Orbit}_G(s)=\{g\cdot s \mid g\in G\}
\end{equation*}
Section 5.3 Orbits and Stabilizers
Subsection In a Stable Orbit
“Being born in a stable does not make one a horse.”―Arthur Wellesley
The information encoded in a group action has two basic parts: one part tells us where points go and the other part tells us how points stay put. The following terminology refers to these ideas.
A fun fact about orbits is that they’ve secretly been at the heart of everything we’ve been doing so far.
Example 5.24. Familiar Orbits.
Let \(G\) be a group.
All the way back in Theorem 3.5 we proved that the set of cosets induced an equivalence relation on the group. As we saw in Example 5.24, left cosets turn out to be a special kind of orbit. Luckily for us, the equivalence relation generalizes.
Definition 5.25.
Let \(G\) be a group acting on a set \(S\text{.}\) The equivalence relation on \(S\) induced by the action of \(G\text{,}\) written \(\sim_G\text{,}\) is defined by \(s\sim_G s'\) if and only if there is a \(g \in G\) such that \(s'=g\cdot s\text{.}\) The equivalence classes of \(\sim_G\) are the orbits.
Theorem 5.26. Orbits Partition the Group.
Let \(G\) be a group acting on a set \(S\text{.}\) Then
- \(\sim_G\) is an equivalence relation;
- for any \(s,s'\in S\text{,}\) either \(\operatorname{Orbit}_G(s)=\operatorname{Orbit}_G(s')\) or \(\operatorname{Orbit}_G(s)\cap \operatorname{Orbit}_G(s')=\emptyset\text{;}\) and
- \(S=\bigcup_{\text{pairwise disjoint orbits}} \operatorname{Orbit}_G(s)\text{.}\)
We’ll have a lot more to say about orbits, but first they’ll need a friend.
Definition 5.27. Stabilizer.
Let \(G\) be a group acting on a set \(X\text{.}\) For an element \(s\in X\) the stabilizer is
\begin{equation*}
\operatorname{Stab}_G(s)=\{g\in G \mid g\cdot s=s\}.
\end{equation*}
Exercise 5.28. Stabilizers are Subgroups.
The stabilizer \(\operatorname{Stab}_G(s)\) is a subgroup of \(G\text{.}\)
One of the most important facts about the action of a group on a finite set is the following:
Theorem 5.29. The Length of the Orbit is the Index of the Stabilizer (LOIS).
Let \(G\) be a group that acts on a finite set \(S\) via \(\cdot\text{.}\) For any \(s \in S\) we have
\begin{equation*}
|\operatorname{Orbit}_G(s)| = [G: \operatorname{Stab}_G(s)]
\end{equation*}
Proof.
Let \(\mathcal L\) be the collection of left cosets of \(\operatorname{Stab}_G(s)\) in \(G\text{.}\) Define a function
\begin{equation*}
\alpha: \mathcal L\to \operatorname{Orbit}_G(s)
\end{equation*}
by \(\alpha(x \operatorname{Stab}_G(s)) = x \cdot s\text{.}\) This function is well defined and one-to-one:
\begin{equation*}
x \operatorname{Stab}_G(s) = y \operatorname{Stab}_G(s)\iff x^{-1}y \in \operatorname{Stab}_G(s) \iff x^{-1}y \cdot s = s \iff y \cdot s = x \cdot s.
\end{equation*}
The function \(\alpha\) is onto by definition of \(\operatorname{Orbit}_G(s)\text{.}\) Thus \(\alpha\) is a bijection and it yields equalities
\begin{equation*}
[G: \operatorname{Stab}_G(s)]=|\mathcal L|=|\operatorname{Orbit}_G(s)|.
\end{equation*}
Length isn’t a word we have used to describe size or cardinality before, but I suppose that SOIS and COIS just don’t quite roll of the tongue quite the same.
Theorem 5.30. Orbit-Stabilizer Theorem.
Let \(G\) be a finite group that acts on a finite set \(S\) via \(\cdot\text{.}\) For any \(s \in S\) we have
\begin{equation*}
|G|=|\operatorname{Orbit}_G(s)|\cdot |\operatorname{Stab}_G(s)|.
\end{equation*}
Warning 5.31.
Take note that Orbit-Stabilizer Theorem only applies when \(G\) is a finite group.
Corollary 5.32. Transitive iff One Orbit.
An action is transitive if and only if there is exactly one orbit under the action. Similarly, an action is transitive if and only if every stabilizer is trivial.
Lets look at some group actions of a more geometric flavor.
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What do you think geometry tastes like? Which field of mathematics would taste the best?
Example 5.33. Symmetries of the Cube.
Let \(G\) be the group of rotational symmetries of the cube and let \(G\) act on the collection of \(6\) faces of the cube in the evident way. This action is transitive and so the one and only orbit has length \(6\text{.}\) It follows that for any face \(f\) of the cube, \(G_f\) has index \(6\) and, since we already know that \(|G|= 24\text{,}\) it follows from Theorem 5.29 that \(|G|_f = 4\text{.}\) That is, there are four symmetries that map \(f\) to itself. Indeed, they are the \(4\) rotations by \(0\text{,}\) \(90\text{,}\) \(180\) or \(270\) degrees about the line of symmetry passing through the mid-point of \(f\) and the mid-point of the opposite face.
That was fun and all, but I just feel like there weren’t enough faces...
Example 5.34. Symmetries of the Dodecahedron.
Let \(G\) be the group of rotatoinal symmetries of the dodecahedron (\(12\) pentagonal faces). The evident action of \(G\) on the twleve faces is transitive. For each face \(f\text{,}\) \(G_f\) clearly has \(5\) elements: the five rotations about the line joining the midpoint of \(f\) to the midpoint of the face on the opposite side. Since \(12 = [G: G_f]\) and \(|G|_f = 5\text{,}\) we conclude \(|G| = 60\text{.}\)
As a final note, we are now able to connect the concepts of transitive and faithful actions in a more meaningful way.
Theorem 5.35. Transitive and Faithful.
Suppose that \(G\) is an abelian group acting transitively and faithfully on a set \(X\text{.}\) Then \(|G|=|X|\text{.}\)
Proof.
Let \(u \in X\text{.}\) As the action is transitive, by the exercise above we have that the kernel of the action is the intersection of \(gG_ug\inv\) over all \(g \in G\text{.}\) However, as \(G\) is abelian, \(gG_ug\inv = G_u\) for all \(g \in G\text{.}\) Hence, the kernel of the action is \(G_u\) for any \(u \in X\text{.}\) On the other hand, the action is faithful, which means the kernel of the action is the identity. It follows that \(G_u = \{e\}\) for every \(u \in X\text{.}\) Finally, as the action is transitive, \(X = \Orb(u)\) for some (any) \(u \in X\text{.}\) By Orbit-Stabilizer Theorem, \(|X|= |\Orb(u)|= [G : G_u] = |G|\text{.}\)
Subsection The Class Equation
“Let the others have the charisma. I’ve got the class.”―George H. W. Bush
Definition 5.36. Centralizer.
Let \(G\) be a group. For any \(g \in G\text{,}\)
\begin{equation*}
Z_G(g):= \{g \in G \mid gx = xg\}
\end{equation*}
is called the centralizer of \(g\) in \(G\)
Exercise 5.37. Center is Intersection of Centrilizers.
Prove that
\begin{equation*}
Z(G)=\bigcap_{g\in G}Z_G(g)
\end{equation*}
Definition 5.38. Normalizer.
Let \(G\) be a group. For any \(S \subseteq G\text{,}\)
\begin{equation*}
N_G(S):= \{g \in G \mid gSg^{-1}=S\}
\end{equation*}
is called the normalizer of \(S\) in \(G\text{.}\)
Remark 5.39.
Notice that when \(S\) consists of just one element, \(x\text{,}\) we have \(N_G(S)=C_G(x)\text{.}\)
Notation Check.
There have been a lot of new sets introduced recently, all of whom seem to be flying around with reckless abandon. Here’s a quick refresher:
Let \(G\) be a group acting on a set \(S\text{,}\) and let \(g\in G, s\in S\text{.}\)
Lemma 5.40. Conjugation Actions and LOIS.
Let \(G\) be a group.
- Then \(G\) acts on \(G\) by Conjugation. For all \(g \in G\text{,}\) the orbit of \(g\) is the conjugacy class of \(g\text{,}\) \(\operatorname{Stab}_G(g)=Z_G(G)\) and \(|C_G(g)| = [G : Z_G(g)]\text{.}\)
- Then \(G\) acts on the power set \(P(G)=\{S\mid S\subseteq G\}\) by Conjugation. For all \(S \in P(G)\text{,}\) \(\operatorname{Stab}_G(S)=N_G(S)\) and \(|\operatorname{Orbit}_G(S)| = [G : N_G(S)]\text{.}\)
Theorem 5.41. Size of Conjugacy Class Divides \(|G|\).
For a finite group \(G\text{,}\) the size of any conjugacy class divides \(|G|\text{.}\)
Theorem 5.42. Class Equation.
Let \(G\) be a finite group and let \(g_1,\ldots g_r \in G\) be a list of unique representatives of all of the conjugacy classes of \(G\) of size greater than \(1\text{.}\) Then
\begin{equation*}
|G| = |Z(G)| + \sum_i^r [G : Z_G(g_i)].
\end{equation*}
Proof.
The elements of \(Z(G)\) are precisely the group elements that are conjugate to only themselves; that is, they are the one-element orbits for the conjugation action. Because the conjugacy classes (orbits of the conjugation action) partition \(G\) we have
\begin{equation*}
|G| = |Z(G)| + \sum_i^r C_G(g_i).
\end{equation*}
For each \(g_i\) as in the statement, by Lemma 5.40, we have \([G: Z_G(g_i)] =C_G(g_i)\text{.}\) The Class Equation follows from substituting this into the equation above.
The Class Equation provides a way of decomposing a group into its conjugacy classes, which are fundamental building blocks of the group and are essential in more advanced topics, such as representation and character theory.
It also gives us information about the size and structure of the center of a group. (Read: when a proof involves the center of a group from here on out, the Class Equation is probably a good place to start)
Summary
- In a group action, the Orbit of an element \(g\) is the set of points that the element is mapped to under the action, and the Stabilizer is set of all points that fix \(g\text{.}\)
- Orbits induce an equivalence relation and partition on \(G\text{.}\)They correspond to cosets under the Left Regular Action and conjugacy classes under the Self Conjugation Action.
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See: Definition 5.25 and Theorem 5.263
See: Example 5.24. - Theorem 5.29, (LOIS), tells us \(|\operatorname{Orbit}_G(s)| = [G: \operatorname{Stab}_G(s)]\) and the Orbit-Stabilizer Theorem yields \(|G|=|\operatorname{Orbit}_G(s)|\cdot |\operatorname{Stab}_G(s)|\text{.}\)
- An action is transitive if and only if there is exactly one orbit under the action. Similarly, an action is transitive if and only if every stabilizer is trivial.
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See: Corollary 5.32. - The Centralizer of an element is the set of all elements that commute with it. The Normalizer of a set is the set of elements that fix the set under conjugation.
- When \(G\) acts on itself by conjugation, we have \(\Orb(g)=C_G(g)\) and \(\operatorname{Stab}_G(g)=Z_G(G)\text{.}\)
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See: Lemma 5.40 - The Class Equation yields \(|G| = |Z(G)| + \sum_i^r [G : Z_G(g_i)]\text{,}\) where \(g_1,\ldots g_r \in G\) are the list of unique representatives of all of the conjugacy classes of \(G\) of size greater than \(1\text{.}\)
