conjugacy classes on symmetric groups
dear math help forums,
i fear that with the advent of my upcoming abstract algebra exam, the world shall end. however, in order to fend off the end of the world, i shall try my best to defeat this exam, and save the planet. your help in my triumph over this exam will be greatly appreciated!
i'm struggling with the following problem:
Find all the conjugacy classes of elements in the symmetric group S4.
i know that since S4 is not abelian, the conjugacy classes will not be singletons. wikipedia tells me the following:
The symmetric group S4, consisting of all 24 permutations of four elements, has five conjugacy classes, listed with their orders:
- no change (1)
- interchanging two (6)
- a cyclic permutation of three (8)
- a cyclic permutation of all four (6)
- interchanging two, and also the other two (3)
however, the numbers after these statements are mystifying to me. any help in explaining how these numbers came into fruition would be crucial to saving our home Earth.