Almost simple groups as flag-transitive automorphism groups of symmetric designs with λ prime

Abstract

In this article, we study symmetric designs with λ prime admitting a flag-transitive and point-primitive automorphism group G of almost simple type with socle X. We prove that either D is one of the six well-known examples of biplanes and triplanes, or D is the pointhyperplane design of PG(n−1, q) with λ = (q n−2 −1)/(q −1) prime and X = PSL n (q).

Reduction for flag‐transitive point‐primitive 2‐(v, k, λ) designs with λ prime

“…In 2020, reduction concerning symmetric designs with $$\lambda $$ prime was studied in [16]. Recently, under the same condition, the classification of automorphism groups with simple socle was finished in [1]. Hence the present paper extends the results from symmetric designs to nonsymmetric 2‐designs.…”

“…Since ≤ ≀ G H S 2 , the flag-transitivity of G implies that ω r 2 divides   H 2 2 , which is impossible for all cases except that z ω λ ( , , ) is (1, 9, 7), (1, 21, 5), (1,33,31) and (4, 21, 5). The four exceptions are ruled out by using the computer software MAGMA [2] with exactly the same procedure in [15,Lemma 3.6], which contains the following four steps:…”

“…But for each ω, we see that b k , or r is not an integer, a contradiction. Now, we work through another case where  λ e, and see that only 1) with ≥ d 5 and ≥ q 4 when T d q = PSL( , ) or d q PSU( , ), respectively. But Lemma 2.3 forces…”

“…Then, similar to the case m = 3 in Lemma 3.1, only T d q = PSL( , ) or d q PSU( , ) is possible. Moreover,  λ e, where ∓ e d q = ( , 1) when T is d q PSL( , ) and d q PSU( , ), respectively. When T is d q PSL( , ), the fact that ≥ λ 5 means ≥ d 5 and ≥ q 7.…”

On Flag-Transitive, Point-Quasiprimitive Symmetric 2-(v,k,λ) Designs with λ Prime

“…In [1], applying the O'Nan-Scott theorem, the first author and Zhou showed that flag-transitive, point-primitive automorphism groups of such symmetric designs have either an abelian socle or a nonabelian simple socle. Alavi et al, in [2], further completed the classification of these designs when the automorphism group has a non-abelian simple socle. In 1993, Praeger established an analog of the O'Nan-Scott theorem for quasiprimitive groups in [3], which implies the possibility of determining the socle of the point-quasiprimitive automorphism group of 2-designs.…”