# Classification of ambivalent alternating groups

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

## Contents

## Statement

The alternating group is an ambivalent group for precisely the following choices of : .

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 for which is strongly ambivalent is precisely the same: .

## Related facts

### Related facts about alternating groups

- Classification of alternating groups having a class-inverting automorphism: This turns out to be .
- Alternating group implies every element is automorphic to its inverse
- Finitary alternating group on infinite set is ambivalent

### Related facts about symmetric groups

- Symmetric groups are rational
- Symmetric groups are rational-representation
- Symmetric groups are ambivalent

### General information pages

## Facts used

## Proof

By fact (1), a product of cycles of distinct odd lengths is conjugate to its inverse if and only if is even. Equivalently, it is conjugate to its inverse if and only if the number of s that are congruent to modulo 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

Thus, the problem reduces to the following: for what can we write in such a way that all are distinct, and the number of that are congruent to modulo is odd? These are precisely the for which is *not* ambivalent.

We quickly see the following:

- can be written in this form, because we can take .
- can be written in this form, because we can take .
- can be written in this form, because we can take .
- can be written in this form, because we can take .

The only cases left are , and it is readily seen that a decomposition into of the above form is not possible for these .