# Regular orbits of symmetric and alternating groups

## TL;DR

This paper classifies when symmetric and alternating groups, via their covering groups, have regular orbits on certain modules over prime order fields, linking to primitive permutation groups of affine type.

## Contribution

It provides a complete classification of pairs (G,V) where G is a covering group of symmetric or alternating groups and has a regular orbit on V over prime order fields.

## Key findings

- Identifies all such pairs (G,V) with regular orbits.
- Connects the problem to primitive permutation groups of affine type.
- Provides criteria for the existence of regular orbits in these cases.

## Abstract

Given a finite group $G$ and a faithful irreducible $FG$-module $V$ where $F$ has prime order, does $G$ have a regular orbit on $V$? This problem is equivalent to determining which primitive permutation groups of affine type have a base of size 2. In this paper, we classify the pairs $(G,V)$ for which $G$ has a regular orbit on $V$ where $G$ is a covering group of a symmetric or alternating group and $V$ is a faithful irreducible $FG$-module such that the order of $F$ is prime and divides the order of $G$.