A note on extremely primitive affine groups

Abstract

AbstractLet G be a finite primitive permutation group on a set $$\Omega $$ Ω with non-trivial point stabilizer $$G_{\alpha }$$ G α . We say that G is extremely primitive if $$G_{\alpha }$$ G α acts primitively on each of its orbits in $$\Omega {\setminus } \{\alpha \}$$ Ω \ { α } . In earlier work, Mann, Praeger, and Seress have proved that every extremely primitive group is either almost simple or of affine type and they have classified the affine groups up to the possibility of at most finitely many exceptions. More recently, the almost simple extremely primitive groups have been completely determined. If one assumes Wall’s conjecture on the number of maximal subgroups of almost simple groups, then the results of Mann et al. show that it just remains to eliminate an explicit list of affine groups in order to complete the classification of the extremely primitive groups. Mann et al. have conjectured that none of these affine candidates are extremely primitive and our main result confirms this conjecture.