# 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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Parents:

• Alternating group

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