Conjugacy classes of the alternating group on five elements

https://arbital.com/p/alternating_group_five_conjugacy_classes

by Patrick Stevens Jun 18 2016 updated Jun 18 2016

$A_5$ has easily-characterised conjugacy classes, based on a rather surprising theorem about when conjugacy classes in the symmetric group split.

This page lists the conjugacy classes of the Alternating group $A_5$ on five elements. See a different lens for a derivation of this result using less theory.

$A_5$ has size $5!/2 = 60$, where the exclamation mark denotes the Factorial function. We will assume access to [4bk the conjugacy class table of $S_5$] the Symmetric group on five elements; $A_5$ is a quotient of $S_5$ by the sign homomorphism.

We have that a conjugacy class splits if and only if its cycle type is all odd, all distinct. (Proof.) This makes the classification of conjugacy classes very easy.

The table

We must remove all the lines of [4bk $S_5$'s table] which correspond to odd permutations (that is, those which are the product of odd-many transpositions). Indeed, those lines are classes which are not even in $A_5$.

We are left with cycle types $(5)$, $(3, 1, 1)$, $(2, 2, 1)$, $(1,1,1,1,1)$. Only the $(5)$ cycle type can split into two, by the splitting condition. It splits into the class containing $(12345)$ and the class which is $(12345)$ conjugated by odd permutations in $S_5$. A representative for that latter class is $(12)(12345)(12)^{-1} = (21345)$.

$$\begin{array}{|c|c|c|c|} \hline \text{Representative}& \text{Size of class} & \text{Cycle type} & \text{Order of element} \\ \hline (12345) & 12 & 5 & 5 \\ \hline (21345) & 12 & 5 & 5 \\ \hline (123) & 20 & 3,1,1 & 3 \\ \hline (12)(34) & 15 & 2,2,1 & 2 \\ \hline e & 1 & 1,1,1,1,1 & 1 \\ \hline \end{array}$$