Characterization of families of rank 3 permutation groups by the subdegrees I

The conditions on N can be replaced by the assumption tha t N is at least 6 and the intersection number/~ has the value ( q § 1) 2. The cases N----4 and 5 are unsettled even under the assumption tha t /~ has this value. We number the main theorems of this sequences of papers consecutively, Theorems I and I I having appeared in [4]. Theorem I I I can be regarded as a linear analogue of Theorem I I , which in turn corresponds to the case q---1 of Theorem I I I . The subgroups of P F L N (q) transitive on the set of 4-simpliees in PN-1 (q) can be regarded as linear analogues of 4-fold transitive permutat ion groups of degTee N. I t does not appear to be known whether PSLN (q) is always the minimal such group. Since this paper was submitted, D. PE~R~ has proved tha t a subgroup of PTLN (q), q 9 2, which is transitive on the set of k-dimensional linear subvarieties of PN-I(q) for some/c, 2 ~ / c _< [2//2] 1, contains PSLN(q). This implies in particular tha t a subgroup of PI'LN(q), N _~ 6, q :~ 2, which has rank 3 on the set of lines