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In 1984, Camina and Gagen gave the result that the block-transitive automorphism group G of a 2-v k ( , , 1)design with  k v must be flag-transitive, moreover, G is point-primitive of affine or almost simple type. As a generalization of this result, the purpose of this paper is to study block-transitive automorphism groups of nontrivial 2-v k λ ( , , ) designs with r k ( , ) = 1, where r is the number of blocks incident with a given point. We prove

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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].