Abstract
Let G be a finite primitive permutation group on a set Omega with non-trivial point stabilizer G_{alpha }. We say that G is extremely primitive if G_{alpha } acts primitively on each of its orbits in Omega {setminus } {alpha }. 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.
Highlights
Let G Sym(Ω) be a finite primitive permutation group with point stabilizer H = Gα = 1
A basic tool in the analysis of these groups is [18, Lemma 4.1], which states that G is not extremely primitive if |M(H)| < 2d/2, where M(H) is the set of maximal subgroups of H
With the exception of the case in the first row, H0 is a simple group of Lie type over F2 and V = L(λ) is a 2-restricted irreducible module for H0 with highest weight λ
Summary
A basic tool in the analysis of these groups is [18, Lemma 4.1], which states that G is not extremely primitive if |M(H)| < 2d/2, where M(H) is the set of maximal subgroups of H. With the exception of the case in the first row, H0 is a simple group of Lie type over F2 and V = L(λ) is a 2-restricted irreducible module for H0 with highest weight λ.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
