# Non-Nudgable Subgroups of Permutations

## TL;DR

This paper investigates subgroups of permutation groups with a symmetry property called non-nudgability, characterizing which groups are non-nudgable or nudgable, and constructing the smallest nudgable group.

## Contribution

It provides a classification of non-nudgable subgroups within permutation groups and constructs the smallest nudgable subgroup, advancing understanding of symmetry properties in permutation groups.

## Key findings

- Full permutation groups are non-nudgable
- Half of the alternating groups are nudgable
- A smallest nudgable subgroup has 6 elements

## Abstract

Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is called non-nudgable, if the cardinality of this set always remains the same when replacing the initial permutation with its inverse. It is called nudgable otherwise. We show that all full permutation groups, standard dihedral groups, half of the alternating groups, and any abelian subgroup are non-nudgable. In the right probabilistic sense, it is thus quite likely that a randomly generated subgroup is non-nudgable. However, the other half of the alternating groups are nudgable. We also construct a smallest possible nudgable group, a 6-element subgroup of the permutation group on 4 elements.