On the rank two geometries of the groups PSL(2, q): part I

Abstract

We determine all firm and residually connected rank 2 geometries on which PSL(2, q) acts flag-transitively, residually weakly primitively and locally two-transitively, where one of the maximal parabolic subgroups is isomorphic to E q : (q−1) (2,q−1) , where E q denotes an elementary abelian group of order q, or D 2n(q) , the dihedral group of order 2n(q) where n(q) := (q±1) gcd(2,q−1) for some prime-power q.

Polarity graphs revisited

“…We will show that PSL(2, q) admits no such proper subgroup H 0 if n ≥ 2, using Dickson's classification of subgroups of PSL(2, q) for q = 2 n as displayed in [13]. By this classification, subgroups of PSL(2, q) split into four classes: (a) cyclic and dihedral subgroups, (b) affine subgroups, (c) subgroups isomorphic to A 4 or A 5 for n even, and (d) subgroups of the form PSL(2, 2 m ) where m is a divisor of n. We analyse the cases separately, recalling that throughout we assume q = 2 n and n ≥ 2.…”

Locally arc-transitive graphs of valence {3,4} with trivial edge kernel

“…Note that the incidence graph of this geometry is precisely the coset graph Cos (G, G 0 , G 1 ). Coset geometries of rank 2 and the corresponding incidence structures have received a considerable attention in the last few years (see [10,11], for example). The results of this paper can be easily interpreted in the language of that particular field of mathematics.…”

On the rank two geometries of the groups PSL(2, q): part II

“…In [5], we started the classification of the residually weakly primitive and locally twotransitive coset geometries of rank two for the groups PSL(2, q). The aim of this paper is to finish this classification.…”

“…It remains to focus on the cases in which one of the maximal parabolic subgroups is isomorphic to A 4 , S 4 , A 5 , PSL(2, q ) or PGL(2, q ) where q divides q. For motivation, basic definitions, notations and context of the work we refer to [5].…”