Nonsymmetric 2‐(v,k,λ) designs, with (r,λ)  = 1, admitting a solvable flag‐transitive automorphism group of affine type

Abstract

Nonsymmetric 2‐(v,k,λ) designs, with (r,λ)=1, admitting a solvable flag‐transitive automorphism group of affine type not contained in AnormalΓL1(v) are classified.

Flag-Transitive Non-Symmetric 2-Designs with $(r, \lambda)=1$ and Exceptional Groups of Lie Type

“…But in each case, r | (|G α |, v − 1) forces r 2 < v, a contradiction. If rank(T 0 ) 1 2 rank(T ), then from Lemma 12, we have |G α | < 4q 20 log p q. By further checking the orders of groups of Lie type, we find that if |G α | < 4q 20 log p q, then |G α | p < q 12 , and so |G α ||G α | 2 p < |G|, contrary to Lemma 9.…”

“…Let G α be maximal in G and the socle T 0 (q) of G α be a simple group of Lie type over F q (q > 2). If rank(T 0 ) 1 2 rank(T ), we have the following bounds:…”

“…Assume that T = E 7 (q). If rank(T 0 ) 1 2 rank(T ), then by Lemma 12 |G α | 3 |G|, a contradiction. If rank(T 0 ) > 1 2 rank(T ), then when q 0 > 2, B by Lemma 11, the only cases G α ∩ T satisfying |G| < |G α | 3 are G α ∩ T = E 7 (q 1 s ), where s = 2 or 3.…”

“…If rank(T 0 ) 1 2 rank(T ), then by Lemma 12 |G α | 3 |G|, a contradiction. If rank(T 0 ) > 1 2 rank(T ), then when q 0 > 2, B by Lemma 11, the only cases G α ∩ T satisfying |G| < |G α | 3 are G α ∩ T = E 7 (q 1 s ), where s = 2 or 3. But in all cases we have r 2 < v. If q 0 = 2, then the possible subgroups T 0 (2) of E 7 (2) such that |G| < |G α | 3 are A 6 (2), A 7 (2), B 5 (2), C 5 (2), D 5 (2) and D 6 (2).…”

“…In [4], P. Dembowski proved that if G is a flag-transitive automorphism group of a 2-design D with (r, λ) = 1, then G is point-primitive and P. H. Zieschang [30] proved that G must be of almost simple or affine type. Such 2-designs have been studied in [1,26,27,28,29], where the socle of G is an elementary abelian p-group, a sporadic group or an alternating group, respectively. In this paper, we continue to study the non-symmetric case that the socle of G is an exceptional simple group of Lie type.…”