Conjugacy Classes
As in the symmetric group, the conjugacy classes in An consist of elements with the same cycle shape. However, if the cycle shape consists only of cycles of odd length with no two cycles the same length, where cycles of length one are included in the cycle type, then there are exactly two conjugacy classes for this cycle shape (Scott 1987, §11.1, p299).
Examples:
- the two permutations (123) and (132) are not conjugates in A3, although they have the same cycle shape, and are therefore conjugate in S3
- the permutation (123)(45678) is not conjugate to its inverse (132)(48765) in A8, although the two permutations have the same cycle shape, so they are conjugate in S8.
Read more about this topic: Alternating Group
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