# Conjugacy classes of the alternating group on five elements

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 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 \(S_5\)’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 \((12)(12345)(12)^{-1} = (21345)\).

Children:

- Conjugacy classes of the alternating group on five elements: Simpler proof
A listing of the conjugacy classes of the alternating group on five letters, without using heavy theory.

Parents:

- Alternating group
The alternating group is the only normal subgroup of the symmetric group (on five or more generators).