= Alternating conjugacy class splitting criterion
{title2=$C_{S_n}(g)\subseteq A_n$}
For an even <permutation> $g$, its <symmetric group> class splits into two equal <alternating group> classes exactly when its symmetric-group centralizer contains no odd permutation. If an odd commuting element exists, an odd conjugator can be multiplied by it to become even; otherwise the index-two subgroup has two conjugation orbits. In cycle notation the splitting types are precisely distinct odd cycle lengths, counting fixed points as length-one cycles: rotating odd cycles is even, whereas an even cycle or the exchange of two equal odd cycles supplies an odd centralizer element.
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