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

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, 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 some prime-power q.

Action of PGL(2,q) on the Cosets of the Centralizer of an Eliptic Element

“…The elements in the sets , and are known as elliptic, parabolic and hyperbolic elements of respectively. (See [2] A summary of the subgroup structure is also found in [3,4] and [5].…”

Rank and subdegrees of PGL(2, q) acting cosets of PGL(2, e) for q an even power of e

“…More details on the subgroup structure of P GL(2, q) and P SL(2, q) are also found in [1], [5], [6] and [10].…”

Two-transitive pairs in $ \mathrm{PSL}(2,q)$

“…The need for the present paper became clear in the study of [4,10] in which the present paper (Theorem 2) was indeed used.…”

“…Our Theorem 1 opens a door to the classification of the rank 3 coset geometries on which G acts flag-transitively. This subject has been preceded by a classification of rank 2 geometries for G in [9], [4] and [10] using a preliminary version of Theorem 2, whose proof refers to the present paper. Theorem 1 also opens a door for a similar treatment in other families of almost simple groups G. There is no need for G to be 2-transitive.…”

“…We take care of delicate arithmetic conditions and we look for classes of configurations that are disjoint. Also, we prepare more complex navigations in the subgroup lattice of PSL(2, q) such as in [4,10].…”