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

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

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  • Alternating group

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