Flag-Transitive Block Designs and Finite Simple Exceptional Groups of Lie Type

On Flag-Transitive $2$-$(k^{2}, k, λ)$ Designs with $λ\mid k$

“…Since then a special attention was given to the case λ > 1. A classification of the flag-transitive 2-designs with gcd(r, λ) = 1, λ > 1 and G AΓL 1 (q), where r is the replication number of D, has been announced by Alavi, Biliotti, Daneshkakh, Montinaro, Zhou and their collaborators in [2] and proven in [3], [4], [5], [8], [10], [11], [30], [37], [39], [40], [41], [42], [44], [45] and [46]. Moreover, recently the flag-transitive 2-designs with λ = 2 have been investigated by Devillers, Liang, Praeger and Xia in [19], where it is shown that apart from the two known symmetric 2-(16, 6, 2) designs, G is primitive of affine or almost simple type.…”

Symmetric designs and projective special unitary groups of dimension at most five

“…In this case, the point-stabiliser is isomorphic to A 5 , see [16,Section 1.2.1]. The complement of this design, which is the one in line 3 of Table 1, is the unique symmetric (11,6,3) design whose full automorphism group PSL (2,11) is also flag-transitive and point-primitive with A 5 as its point-stabiliser, see also [22,Theorem 1.2].…”

“…The unique symmetric (15,8,4) design D in line 4 of Table 1 can be constructed by points and complements of hyperplanes of PG (3,2). The full automorphism group of D is PSL(4, 2) ∼ = A 8 which admits a proper subgroup PSL(2, 9) ∼ = A 6 as an automorphism group of D. Note that PSL(2, 9) acts flag-transitively on D with point-stabiliser S 4 , but not on its complement D * .…”

“…The designs in lines 7-11 in Table 1 also arise from studying rank three permutation groups. The complement of the symmetric (56, 11, 2) design is the design with parameters (56, 45, 18) admitting a rank three automorphism group PSL (3,4) as its socle, see [6] and [11,Theorem].…”

“…Although, it is still unknown for larger λ such an automorphism group is of these two types, it is somehow interesting to study such designs whose automorphism group G is an almost simple group with socle X. This paper is part of contribution in classification of symmetric designs admitting flag-transitive and point-primitive finite almost simple automorphism groups of Lie type of small dimension, see [1,2,3,4,10]. In this paper, we continue this project to obtain symmetric designs admitting an automorphism group whose socle is a projective special unitary group of dimension at most five.…”