See element structure of alternating group:A4 for background information.
For a symmetric group, cycle type determines conjugacy class. The statement is almost true for the alternating group, except for the fact that some conjugacy classes of even permutations in the symmetric group split into two in the alternating group, as per the splitting criterion for conjugacy classes in the alternating group, which says that a conjugacy class of even permutations splits in the alternating group if and only if its cycle decomposition comprises odd cycles of distinct length.
Here are the unsplit conjugacy classes:
|Partition||Verbal description of cycle type||Elements with the cycle type||Size of conjugacy class||Formula for size||Element order|
|1 + 1 + 1 + 1||four cycles of size one each, i.e., four fixed points||-- the identity element||1||1|
|2 + 2||double transposition: two cycles of size two||, ,||3||2|
|Total||--||, , and||4||NA||NA|
In this case, the union of the unsplit conjugacy classes is a proper normal subgroup isomorphic to the Klein four-group. Note that this phenomenon is unique to the case .
Here is the split conjugacy class:
|Partition||Verbal description of cycle type||Elements with the cycle type||Combined size of conjugacy classes||Formula for combined size||Size of each half||First split half||Second split half||Real?||Rational?||Element order|
|3 + 1||one 3-cycle, one fixed point||, , , , , , ,||8||4||, , ,||, , ,||No||No||3|