Flag-transitive block designs and unitary groups
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Flag-transitive block designs and unitary groups Seyed Hassan Alavi1 · Mohsen Bayat1 · Ashraf Daneshkhah1 Received: 29 March 2020 / Accepted: 19 April 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this article, we determine all pairs (D, G), where D is a 2-design with gcd(r , λ) = 1 and G is a flag-transitive almost simple automorphism group of D whose socle is PSU(n, q) with (n, q) = (3, 2). We prove that such a design belongs to one of the two infinite families of Hermitian unitals and Witt–Bose–Shrikhande spaces, or it is isomorphic to a design with parameters (6, 3, 2), (7, 3, 1), (8, 4, 3), (10, 6, 5), (11, 5, 2) or (28, 7, 2). In particular, if D is symmetric, then it is either the Fano plane with parameters (7, 3, 1), or the unique Hadamard design with parameters (11, 5, 2). Keywords 2-Design · Flag-transitive · Automorphism group · Almost simple group · Unitary group · Large subgroup Mathematics Subject Classification 05B05 · 05E18 · 20D05
1 Introduction A 2-design D with parameters (v, k, λ) is a pair (P, B) with a set P of v points and a set B of b blocks such that each block is a k-subset of P and each two distinct points are contained in λ blocks. We say D is nontrivial if 2 < k < v −1, and symmetric if v = b. Each point of D is contained in exactly r blocks which is called the replication number of D. A flag of D is a point-block pair (α, B) such that α ∈ B. An automorphism of a 2-design D is a permutation of the points permuting the blocks and preserving the
Communicated by John S. Wilson.
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Seyed Hassan Alavi [email protected]; [email protected] Mohsen Bayat [email protected] Ashraf Daneshkhah [email protected]
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Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran
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S. H. Alavi et al.
incidence relation. The full automorphism group Aut(D) of D is the group consisting of all automorphisms of D. For G Aut(D), G is called flag-transitive if G acts transitively on the set of flags and G is said to be point-primitive if it is primitive on P. A group G is said to be almost simple with socle X if X G Aut(X ), where X is a nonabelian simple group. We here adopt the standard notation for finite simple groups of Lie type, for example, we use PSL(n, q), PSp(n, q), PSU(n, q) and P (n, q) with ∈ {◦, −, +} to denote the finite classical simple groups. Symmetric and alternating groups on n letters are denoted by Symn and Altn , respectively. A group of order n is simply denoted by n. We use Cn to denote the cyclic group of order n. Further notation and definitions in both design theory and group theory are standard and can be found, for example in [9,14,16]. The main aim of this paper is to study 2-designs with gcd(r , λ) = 1 admitting a flag-transitive automorphism group G. According to [15, 2.3.7], the automorphism group G is point-primitive, and in 1988, Zieschang [34] proved that G is of almost simple or affine type. Such designs admitting an almost simple automorphism group with socle being an alternati
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