# 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 $(12)(12345)(12)^{-1} = (21345)$.

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