# Splitting criterion for conjugacy classes in the alternating group

## Statement

Suppose $n$ is a natural number. Let $S_n$ be the symmetric group on $n$ letters, and $A_n$ be the alternating group: the subgroup of even permutations. Then, given a conjugacy class in $S_n$ of even permutations (i.e., a conjugacy class that lies completely inside $A_n$), the following two possibilities hold:

• This conjugacy class is equal to a single conjugacy class in $A_n$.
• This conjugacy class splits into two conjugacy classes in $A_n$.

The criterion that determines the fate of a conjugacy class (i.e., whether or not it splits) is termed the splitting criterion, and this criterion can be formulated both in the language of centralizers and the language of cycle decompositions.

### In terms of centralizers

The conjugacy class of an element $g \in A_n$:

• splits if the centralizer $C_{S_n}(g)$ of $g$ in $S_n$ is contained in $A_n$, i.e., if any permutation that commutes with $g$ is even; and
• does not split if the centralizer $C_{S_n}(g)$ of $g$ in $S_n$ is not contained in $A_n$, i.e., if there exists an odd permutation that commutes with $g$.

### In terms of cycle decompositions

The conjugacy class of an element $g \in A_n$:

• splits if the cycle decomposition of $g$ comprises cycles of distinct odd length. Note that the fixed points are here treated as cycles of length $1$, so it cannot have more than one fixed point; and
• does not split if the cycle decomposition of $g$ contains an even cycle or contains two cycles of the same length.

## Related facts

### Related facts for symmetric groups on finite sets

Fact Context Statement
Cycle type determines conjugacy class symmetric group on finite set, also symmetric group on infinite set The conjugacy class of an element is determined by and determines the cycle type of the element, i.e., information about the number of cycles of each size in the cycle decomposition of the element.
Classification of ambivalent alternating groups alternating group on finite set The classification of the finite set of values $n$ such that the alternating group $A_n$ is an ambivalent group, i.e., every element is conjugate to its inverse. Ambivalent finite groups are alternately characterized by a character table with all entries real.
Classification of alternating groups having a class-inverting automorphism alternating group on finite set The classification of the finite set of values $n$ such that the alternating group $A_n$ is a group having a class-inverting automorphism: an automorphism that sends every conjugacy class to the conjugacy class of its inverse elements.

## Particular cases

Value of $n$ Number of conjugacy classes in $A_n$ that split Number of elements in these conjugacy classes Partitions of $n$ corresponding to these conjugacy classes Number of conjugacy classes in $A_n$ that don't split Number of elements in these conjugacy classes Partitions of $n$ corresponding to these conjugacy classes Total number of $S_n$-conjugacy classes in $A_n$ Total number of conjugacy classes in $A_n$
2 0 0 -- 1 1 1 + 1 1 1
3 1 2 3 1 1 1 + 1 + 1 2 3
4 1 8 3 + 1 2 4 1 + 1 + 1 + 1, 2 + 2 3 4
5 1 24 5 3 36 1 + 1 + 1 + 1 + 1, 3 + 1 + 1, 2 + 2 + 1 4 5
6 1 144 5 + 1 5 216 1 + 1 + 1 + 1 + 1, 3 + 1 + 1 + 1, 2 + 2 + 1 + 1, 3 + 3, 4 + 2 6 7