Conjugacy-Class Splitting in the Alternating Group
For , its -conjugacy class stays one -class when an odd permutation centralises ; otherwise it splits into two equal -classes.
Let .
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If some odd element of commutes with , then
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Otherwise, if every element of that commutes with is even, then splits into two:
where is any transposition (or any odd permutation).
Proof
Orbit-stabiliser and give
The subgroup is the kernel of the sign homomorphism restricted to , so its index is or . If an odd element centralises , the index is and the two conjugacy classes have equal size, hence are equal. Otherwise the centralisers coincide and the -class is twice the size of the -class. Conjugating by any odd permutation supplies the second half.
Examples
In , the identity and -cycle classes remain intact, while the -cycle class splits into two classes of size .
In , the classes represented by , , and have sizes , , and . The -cycle class splits into two classes of size .
Related
Stated in
- Lemma 8.25 (Conjugacy Classes In )§8.3 Conjugacy in and