Regular subgroups of the affine group with no translations

M. A. Pellegrini, M. C. Tamburini Bellani

Published 2016-08-04Version 1

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 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 construct 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 F to be finite in certain cases.