Review of Group Theory: Group Actions (Pt. II Orbits and the Orbit Decomposition Theorem)
Point of post: In this post we discuss more in-depth the concept of the orbits of a -action and show how it carves up a group into a partition, which for finite groups gives us a lot of information.
In prior posts we’ve discussed Lagrange’s theorem and some of the profound consequences it can have on the study of finite groups. Look closely though and one will see that Lagrange’s theorem was really just a consequence of group actions. Namely, we had was a finite group and then acted naturally on by . One can quick check that the orbits of these actions are the right cosets. The reason why this action was powerful was the way in which it carved up into disjoint subsets, which we could exploit to tell us interesting things combinatorially/number theoretically about the order of . It turns out that this wasn’t a coincidence. In particular, every -action on a set carves up in a particularly nice way. This post will be devoted to studying in which way precisely this is.
Recall that if is a -space with -action then for we define the orbit of to be
and the isotropy subgroup of to be
(also known as the stabilizer subgroup and denoted ). We begin our discussion by showing that every -space can be decomposed by its orbits in a way made precise by the theorem. Namely:
Theorem(Orbit Decomposition Theorem): Let be a -space with -action . Then, the relation
is an equivalence relation on and where is the equivalence class of under . In particular if denotes the set of (distinct) orbits on under then
Proof: To see that is an equivalence relation it suffices, by definition, to show that it’s reflexive, symmetric, and transitive. The first of these is clear since and so . Suppose next that then there exists such that , then and so . Finally, if and then there exists such that and and so and so . It follows that is an equivalence relation.
To see that we merely note that
from where the conclusion follows. This last part follows since being the set of equivalence classes of an equivalence relation forms a partition of .
Corollary: Let be a finite -space then
The problem with the above theorem (ostensibly) is that it may, in most cases, be nigh impossible to calculate . Our next theorem says that this is not the case, in fact calculating is as easy as calculating the index of a series of subgroups of . More precisely:
Theorem(Orbit-Stabilizer Theorem): Let be -space with -action . Then, for each fixed the mapping
is a well-defined bijection (where here just denotes the set of left cosets, it isn’t endowed with a group structure since it’s feasible that is not normal in ).
Proof: The issue for why might not be well-defined is that it’s conceivable that yet and so it’s theoretically possible that . To see that this can’t happen we merely note that if then and thus and therefore
and thus . Moreover, to see that is injective note that if then for some and thus
Noting that is clearly surjective finishes the argument.
Corollary: Let be a finite -space with -action . Then, if is a set of one element from each distinct member of then
and if is finite
1. Lang, Serge. Undergraduate Algebra. 3rd. ed. Springer, 2010. Print.
2. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 200