Conjugacy classes of the alternating group on five elements: Simpler proof

https://arbital.com/p/conjugacy_classes_alternating_five_simpler

by Patrick Stevens Jun 18 2016 updated Jun 18 2016

A listing of the conjugacy classes of the alternating group on five letters, without using heavy theory.


The Alternating group on five elements has conjugacy classes very similar to those of the Symmetric group , but where one of the classes has split in two.

Note that has elements, since it is precisely half of the elements of which has elements (where the exclamation mark is the Factorial function).

Table

Working

Firstly, the identity is in a class of its own, because for every .

Now, by the same reasoning as in the proof that conjugate elements must have the same cycle type in , that result also holds in .

Hence we just need to see whether any of the cycle types comprise more than one conjugacy class.

Recall that the available cycle types are , , , (the last of which is the identity and we have already considered it).

Double-transpositions

All the double-transpositions are conjugate (so the cycle type does not split):

Three-cycles

All the three-cycles are conjugate (so the cycle type does not split):

Five-cycles

This class does split: I claim that and are not conjugate. (Once we have this, then the class must split into two chunks, since is closed under conjugation in , and is closed under conjugation in . The first is the conjugacy class of in ; the second is the conjugacy class of . The only question here was whether they were separate conjugacy classes or whether their union was the conjugacy class.)

Recall that , so we would need a permutation such that sends to , to , to , to , and to . The only such permutation is , the transposition, but that is not actually a member of .

Hence in fact and are not conjugate.