Flag-transitive quasi-symmetric designs with block intersection numbers 0 and 2

Abstract

Quasi-symmetric designs are the special type of [Formula: see text]-designs with two block intersection numbers [Formula: see text] and [Formula: see text]. In this paper, we study the automorphism groups of quasi-symmetric designs with block intersection numbers [Formula: see text] and [Formula: see text]. We prove that for a quasi-symmetric design [Formula: see text] with [Formula: see text] and [Formula: see text] satisfying that [Formula: see text] does not divide [Formula: see text], if [Formula: see text] is flag-transitive, then [Formula: see text] must be point-primitive. We further obtain that [Formula: see text] is either of affine type or almost simple type by the O’Nan Scott Theorem. Moreover, when the socle of [Formula: see text] is sporadic, [Formula: see text] is a unique [Formula: see text]-[Formula: see text] design and [Formula: see text] or [Formula: see text].