Correction to: Flag-transitive block designs and unitary groups

Abstract

In this note, we cover a gap in the proof of [2, Proposition 4.3]. In conclusion, Theorem 1.1 in [2] is revisited: if 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), then D belongs to one of the three infinite families of Hermitian unitals, Witt-Bose-Shrikhande spaces and 2-designs with parameters (q 3

Classification of the non-trivial $2$-$(k^{2},k,λ)$ designs, with $λ\mid k$, admitting a flag-transitive almost simple automorphism group

“…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], [9], [10], [8], [16], [17], [42], [52], [55], [56], [57], [58], [59], [60] and [61]. Moreover, recently the flag-transitive 2-designs with λ = 2 have been investigated by Devillers, Liang, Praeger and Xia in [26], 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.…”

“…Thus i = 1, k 2 = q n −1 q−1 , and hence (n, q, k) = (4, 7, 20) or (5,3,11) again by [49], A8.1. If (n, q, k) = (4,7,20), then b = 21 • 20 • λ with λ | 20 and λ 2. However, P SL 4 (7) G P GL 4 (7) has no transitive permutation representations of degree b by [14], Tables 8.8-8.9.…”

“…If (n, q, k) = (4,7,20), then b = 21 • 20 • λ with λ | 20 and λ 2. However, P SL 4 (7) G P GL 4 (7) has no transitive permutation representations of degree b by [14], Tables 8.8-8.9. So, (n, q, k) = (5,3,11), and hence b = 12 • 11 2 since λ | 11 and λ 2.…”

“…3 3 by Corollary 4.10(1). Then n = 4, q = 7, X is isomorphic to one of the groups P SL 4 (7) or P Sp 4 (7) (see Remark 4.2), and…”

“…If X ∼ = P Sp 4 (7), then b = [X : X B ] = 54880, since A 7 is maximal in P Sp 4 (7), and this one has a unique conjugacy class of subgroups isomorphic to A 7 , by [14], Tables 8 for X ∼ = P Sp n (q), or…”