Regular subgroups of the affine group with no translations
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.
Keywords
Cite
@article{arxiv.1608.02492,
title = {Regular subgroups of the affine group with no translations},
author = {M. A. Pellegrini and M. C. Tamburini Bellani},
journal= {arXiv preprint arXiv:1608.02492},
year = {2017}
}
Comments
With respect to the previous version, some results about characteristic 2 were extended from finite fields to any fields