Flag-transitive block designs and unitary groups

Correction to: Flag-transitive block designs and unitary groups

“…We are now ready to revisit Proposition 4.3 in [2], and prove Proposition 2.1 below. In what follows, we frequently use the results mentioned above about the Hermitian unitals and their automorphism groups.…”

“…An automorphism of a 2-design D is a permutation of the points permuting the blocks and preserving the 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. In this note, we cover a gap in the proof of [2,Proposition 4.3]. Therefore, we correct Theorem 1.1 in [2] as below: Theorem 1.1 Let D be a nontrivial 2-design with gcd(r , λ) = 1, and let α be a point of D. Suppose that G is an automorphism group of D whose socle is X = PSU(n, q) with (n, q) = (3, 2).…”

“…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. In this note, we cover a gap in the proof of [2,Proposition 4.3]. Therefore, we correct Theorem 1.1 in [2] as below: Theorem 1.1 Let D be a nontrivial 2-design with gcd(r , λ) = 1, and let α be a point of D. Suppose that G is an automorphism group of D whose socle is X = PSU(n, q) with (n, q) = (3, 2). If G is flag-transitive, then λ ∈ {1, 2, 3, 5} and v, k, λ, X α and X are as in one of the lines in Table 1 or one of the following holds: It is worth noting by [6] that there is a general construction method for 2-designs from linear space:…”

“…

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

…”

“…Proof Suppose that H = G α with α a point of D. If H is not a parabolic subgroup P m , then we follow the same argument as in [2,Proposition 4.3] which leads to no possible parameters. Therefore, considering Remark 1.1, we only need to deal with the case where H is isomorphic to P m , for some 2m ≤ n. In this case, by the same argument as in [2,Proposition 4.3], the inequality v < r 2 restricts to the case where n = 3, that is to say, X = PSU(3, q) and H ∩ X ∼ =ˆq 3 (q 2 − 1) in which case v = q 3 + 1. If λ = 1, then by [10], D is a Hermitian unital as in part (a).…”