Minimal degree for a permutation representation of a classical group

Large doubly transitive orbits on a line

“…Then the assertion follows by [34,Haupsatz H.8.27] when H = PSL (2, q). Furthermore, the assertion follows by [25] and [49] when H = PSL (3, q [43,Chapter 4], in conjunction with [12] and with Lemma 4 and Table II of [45], shows that the assertion is true also in this case.…”

Quadratic Extensions of Flag-transitive Planes

“…Furthermore, Pentilla and Williams [31] have constructed an infinite class of cyclic regular parallelisms in P G (2, q) for q ≡ 2 mod 3. The previously known regular parallelisms are also cyclic and lie in P G (3,2), P G (3,5) and P G (3,8).…”

“…This means that the group S L(2, q) is generated by elation groups and that G L(2, q) leaves invariant each subplane of order q incident with the zero vector in the associated G F(q)-regulus net defined by the elation axes of S L (2, q). Furthermore, there are always infinite point orbits of lengths q + 1 and q 3 − q, which implies that we have a very large assortment of cubic extensions of a flag-transitive plane.…”

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“…(b) If T is nonabelian then, by [13], T is one of the groups listed in (b). 2, 7), (2, 9), (2, 11), or (4, 2) (see [8], [9] or [11]). Moreover since PSL m-1 (r) has a subgroup of odd prime index q = 3(mod 4), q < (r m-1 -l)/(r-1), (m-1, r) is (2, 7) or (2,11).…”