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Splitting criterion for conjugacy classes in double cover of alternating group

Contents

Statement

Goal of the statement

The goal of the statement is as follows. Consider a natural number n \ge 4. We are interested in the quotient map from the double cover of alternating group 2 \cdot A_n to the alternating group A_n.

For any conjugacy class c of A_n of size r, the inverse image of c in 2 \cdot A_n has size 2r. This inverse image is either a single conjugacy class of size 2r or it splits as a union of two conjugacy classes, each of size r. The goal is to determine a condition on the cycle type of the permutations in the conjugacy class that controls whether or not the inverse image splits.

Explicit condition

Consider a conjugacy class c of A_n of size r and element order d whose cycle type is a given unordered integer partition of n. Note that any conjugacy class in A_n has a unique cycle type. Conversely, for every unordered integer partition with an even number of cycles of even size, there is either one or two conjugacy classes with that cycle type (more on this later). The explicit condition on the unordered integer partition is as follows:

Does the partition have at least one even part? Does the partition have a repeated part? (the repeated part may be even or odd) Conclusion
No irrelevant The inverse image splits in two conjugacy classes, each of size r. One of the conjugacy classes has element order d and the other conjugacy class has element order 2d. Due to the different element orders, the conjugacy classes are not interchanged by outer automorphisms.
Yes No The inverse image splits in two conjugacy classes, each of size r and element orders 2d. The two conjugacy classes are interchanged by an outer automorphism.
Yes Yes The inverse image remains a single conjugacy class of size 2r and element orders 2d.

Condition rewritten in terms of number of conjugacy classes for a given cycle type

Consider an unordered integer partition. If the unordered integer partition has an even number of even parts, then it is the cycle type of an even permutation. For even permutations, we would like to know, first, how many conjugacy classes there are in A_n with that cycle type. This number is either 1 or 2, depending on whether the conjugacy class splits from S_n to A_n. For each such conjugacy class, we would like to know whether its inverse image in 2 \cdot A_n splits.

To answer both these questions, we must combined this splitting criterion with the splitting criterion for conjugacy classes in the alternating group. The overall conclusions are below. REMEMBER THAT EVERYTHING BELOW IS FOR PARTITIONS WITH AN EVEN NUMBER OF EVEN PARTS, i.e., those that correspond to even permutations. Note that the first two columns are hypotheses we make about the partition and the third and fourth columns are conclusions we draw about splitting:

Hypothesis: does the partition have at least one even part? Hypothesis: does the partition have a repeated part? (the repeated part may be even or odd) Conclusion: does the conjugacy class split from S_n to A_n in 2? Conclusion: does the fiber in 2 \cdot A_n over a conjugacy class in A_n split in 2? Total number of conjugacy classes in 2 \cdot A_n corresponding to this partition (4 if Yes to both preceding columns, 2 if Yes to one and No to other, 1 if No to both)
No No Yes Yes 4
No Yes No Yes 2
Yes No No Yes 2
Yes Yes No No 1

Orders of elements

For a given unordered integer partition, the lcm of the parts gives the order of any element of S_n with that partition as its cycle type. If the partition has an even number of even parts, the conjugacy class is in A_n and the order as an element of A_n is the same: the lcm of the parts.

When we take the inverse image in 2 \cdot A_n, the order of elements in the inverse image is either the same or twice the order of the element in A_n. We provide below a slight modification of the previous table that includes order information:

Hypothesis: does the partition have at least one even part? Hypothesis: does the partition have a repeated part? (the repeated part may be even or odd) Conclusion: does the conjugacy class split from S_n to A_n in 2? Conclusion: does the fiber in 2 \cdot A_n over a conjugacy class in A_n split in 2? Total number of conjugacy classes in 2 \cdot A_n corresponding to this partition (4 if Yes to both preceding columns, 2 if Yes to one and No to other, 1 if No to both) Number of these conjugacy classes where order of element = lcm of parts Number of these conjugacy classes where order of element = twice the lcm of parts
No No Yes Yes 4 2 2
No Yes No Yes 2 1 1
Yes No No Yes 2 0 2
Yes Yes No No 1 0 1

Conclusion for number of conjugacy classes in double cover of alternating group

For more on this, see element structure of double cover of alternating group#Combinatorics.