## 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.

*Motivation*

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.

*Orbits*

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*

**References:**

1. Lang, Serge. *Undergraduate Algebra*. 3rd. ed. Springer, 2010. Print.

2. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 200

[…] By the Orbit Decomposition Theorem we know […]

Pingback by Review of Group Theory: Group Actions (Pt. IV Conjugation and the Class Equation) « Abstract Nonsense | January 6, 2011 |

[…] verified to be a group action with the orbit of equal to and isotropy subgroups . Thus, by earlier theorem we know that for every . If we say that is conjugate to . So, with one small lemma we are ready […]

Pingback by Review of Group Theory: Sylow’s Theorems (Pt. II) « Abstract Nonsense | January 8, 2011 |

[…] by the Orbit Decomposition Theorem […]

Pingback by Review of Group Theory: Alternate Proof of the Sylow Theorems « Abstract Nonsense | January 11, 2011 |

[…] but this clearly implies that , which is a contradiction. It follows then that for every with we have that . But, by the Orbit-Stabilizer Theorem […]

Pingback by Review of Group Theory: Alternate Proof of the Sylow Theorems (Pt. II) « Abstract Nonsense | January 12, 2011 |

[…] both sides and recalling by the orbit stabilizer theorem that we see […]

Pingback by Representation Theory: Second Orthogonality Relation For Irreducible Characters « Abstract Nonsense | March 9, 2011 |

[…] By first principles we know that is non-trivial. So, we may choose a non-trivial . From the orbit stabilizer theorem we know that where is the conjugacy class containing and ‘s centralizer. It follows from […]

Pingback by Representation Theory: Burnside’s Theorem « Abstract Nonsense | March 11, 2011 |

[…] to the conjugacy class of since it is the only trivial conjugacy class). Thus, . But, by the orbit-decomposition theorem that and so doing the simple computation we find that . To find we recall (considering the […]

Pingback by University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) (August-2003) « Abstract Nonsense | May 1, 2011 |

[…] Let be a finite group which acts on the finite set , suppose that breaks up into oribts as where is transveral for the set of orbits . Then, if is permutation representation of on […]

Pingback by Permutation Representation « Abstract Nonsense | May 2, 2011 |

[…] by the orbit-stabilizer theorem we have that . What we claim though is that . Indeed, if then for some and so and thus […]

Pingback by Double Cosets « Abstract Nonsense | May 2, 2011 |

[…] We use the orbit-stabilizer theorem we write […]

Pingback by Composition of the Restriction Map and the Induction Map « Abstract Nonsense | May 4, 2011 |

[…] commutes with every element of and since we have by b) that . Recall though that by the orbit-stabilizer theorem that and since we have that for some , and so if we must have that […]

Pingback by University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) (January-2004) (Pt. I) « Abstract Nonsense | May 6, 2011 |

[…] and the associated orbit under the action of on induced by . Then, is […]

Pingback by Representation Theory of Semidirect Products: The Preliminaries (Pt. III) « Abstract Nonsense | May 8, 2011 |

[…] We thus see that must map a -cycle to a -cycle for which there are ways to do. Thus, if we fix a particular permutation of the -cycles then choosing a particular -cycle we see that must only cyclically permute the elements of this cycle, for which there are obviously ways to do, and thus within each fixed choice of permutations of the -cycles there are choices for ‘s action. Thus, for each there are choices and thus the total number of choices for is . Thus, we may conclude that from where the conclusion follows by the orbit-stabilizer theorem. […]

Pingback by Conjugacy Classes on the Symmetric Group « Abstract Nonsense | May 10, 2011 |

[…] for the action of on . We can obviously partition into the two sets of all with and . From the Orbit Decomposition Theorem we know […]

Pingback by Actions by p-Groups (Pt. I) « Abstract Nonsense | September 15, 2011 |