Regular subgroups of the affine group with no translations

TL;DR

This paper investigates the existence of regular subgroups of the affine group that contain no translations, providing new examples for all dimensions n ≥ 5 and establishing conditions for smaller n.

Contribution

It constructs examples of regular subgroups with no translations in AGL_n(F) for all n ≥ 5 and characterizes their existence for n < 5 under unipotency assumptions.

Findings

Examples of such subgroups are given for all n ≥ 5.

Necessary and sufficient conditions are established for n < 5.

The work extends previous results by generalizing to any field F.

Abstract

Given a regular subgroup R of AGL_n(F), one can ask if R contains nontrivial translations. A negative answer to this question was given by Liebeck, Praeger and Saxl for AGL_2(p) (p a prime), AGL_3(p) (p odd) and for AGL_4(2). A positive answer was given by Hegedus for AGL_n(p) when n >= 4 if p is odd and for n=3 or n >= 5 if p=2. A first generalization to finite fields of Hegedus' construction was recently obtained by Catino, Colazzo and Stefanelli. In this paper we give examples of such subgroups in AGL_n(F) for any n >= 5 and any field F. For n < 5 we provide necessary and sufficient conditions for their existence, assuming R to be unipotent if char F=0.