Classification of ambivalent alternating groups

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This article classifies the members in a particular group family alternating group that satisfy the group property ambivalent group.

Statement

The alternating group A_n is an ambivalent group for precisely the following choices of n: n = 1,2,5,6,10,14.

Note that in the proof, we show that in each of these cases, for every element in the alternating group, there is an element of order two in the alternating group conjugating it to its inverse. Thus, we show that the set of n for which A_n is strongly ambivalent is precisely the same: n = 1,2,5,6,10,14.

Related facts

Related facts about alternating groups

Related facts about symmetric groups

General information pages

Facts used

  1. Criterion for element of alternating group to be real

Proof

By fact (1), a product of cycles of distinct odd lengths r_1,r_2,\dots,r_k is conjugate to its inverse if and only if \sum (r_i - 1)/2 is even. Equivalently, it is conjugate to its inverse if and only if the number of r_is that are congruent to 3 modulo 4 is even. Note also that if it is conjugate to its inverse, we can choose as our conjugating element an element of order two: the product of transpositions described above.

What this boils down to for n

Thus, the problem reduces to the following: for what n can we write n = \sum_{i=1}^k r_i in such a way that all r_i are distinct, and the number of r_i that are congruent to 3 modulo 4 is odd? These are precisely the n for which A_n is not ambivalent.

We quickly see the following:

  • n = 4d + 3 can be written in this form, because we can take k = 1, r_1 = 4d + 3.
  • n = 4d + 4 can be written in this form, because we can take k = 2, r_1 = 4d + 3, r_2 = 1.
  • n = 4d + 9 can be written in this form, because we can take k = 3, r_1 = 1, r_2 = 5, r_3 = 4d + 3.
  • n = 4d + 18 can be written in this form, because we can take k = 4, r_1 = 1, r_2 = 5, r_3 = 9, r_4 = 4d + 3.

The only cases left are n = 1,2,5,6,10,14, and it is readily seen that a decomposition into r_i of the above form is not possible for these n.