Orbits of $Z \circ (2.O_8^+(2).2)$ in Dimension 8

TL;DR

This paper determines when certain group actions on 8-dimensional modules over finite fields have regular orbits, focusing on groups related to $2.O_8^+(2)$ and their scalar subgroups over primes greater than 7.

Contribution

It explicitly identifies primes and subgroups for which the central products have regular orbits on the module, clarifying an omission in previous work related to the $k(GV)$-problem.

Findings

Identifies primes $p > 7$ where regular orbits exist.

Determines which subgroups $Z$ act with regular orbits.

Completes the analysis missing in prior literature.

Abstract

Groups of structure $2.O_8^+(2)$ have an irreducible representation of degree $8$ which can be realized over $\mathbb{Z}$ and any prime field $\mathbb{F}_p$. This representation extends to a group of structure $2.O_8^+(2).2$. Any subgroup $Z \leq \mathbb{F}_p^{\times}$ acts by scalar multiplication on this module over $\mathbb{F}_p$. In this short note we determine for which primes $p > 7$ and which $Z$ the central products $Z \circ (2.O_8^+(2)$ and $Z \circ (2.O_8^+(2).2)$ have a regular orbit on the $8$-dimensional $\mathbb{F}_p$-module. This work was triggered by an omission in the paper by K\"ohler and Pahlings with title 'Regular Orbits and the $k(GV)$-Problem', a paper which is used in various places in work on the $k(GV)$-problem.