Let G=S_n and g an element of A_n. The conjugacy class of g in G splits into 2 classes in A_n if and only if the centralizer C_G(g) is contained in A_n.
The more concrete criteria is: the conjugacy class of g in G splits if and only if g is the product of cycles of distinct odd length.
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.
e.g. The two permutations (123) and (132) are not conjugates in A3, although they have the same cycle shape, and are therefore conjugate in S3
Thanks, but in fact I hope to get example of even permutation which is a product of disjoint odd cycles as Dr Wang has mentioned.
Dear Amr,
A given permutation is either a product of an even number of transpositions or a
product of an odd number of transpositions, but never both.
A permutation is even if it is expressible as a product of an even number of transpositionsand odd if it is a product of an odd number of transpositions.
Thanks,Dr Wang , could you refer me to a reference related to the matter
Hi Amr, a conjugacy class of the symmetric group Sn separates into 2 classes inside the alternating group An if the cycle decomposition of any element contained in that conjugacy class comprises of cycles of distinct odd length.
As a consequence of this, for every odd n, the conjugacy class consisting of elements of cycle length n, separates into two classes in An (although not unique, consider A15, 15 = 15 = 3+5+7).
Other simple examples are conjugacy classes of element of cycle-type 1,3 in A4 , 5 in A5, 1,5 in A6, etc.
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