Editing Alternating group (section)

== Conjugacy classes ==
As in the [[symmetric group]], any two elements of A<sub>''n''</sub> that are conjugate by an element of  A<sub>''n''</sub> must have the same [[cycle shape]]. The converse is not necessarily true, 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 {{harv|Scott|1987|loc=§11.1, p299}}.

Examples:
*The two [[permutation]]s (123) and (132) are not conjugates in A<sub>3</sub>, although they have the same cycle shape, and are therefore conjugate in S<sub>3</sub>.
*The permutation (123)(45678) is not conjugate to its inverse (132)(48765) in A<sub>8</sub>, although the two permutations have the same cycle shape, so they are conjugate in S<sub>8</sub>.