On Flag-Transitive $2$-$(k^{2}, k, λ)$ Designs with $λ\mid k$

Abstract

It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group G of a 2-(k 2 , k, λ) design D, with λ | k, is either an affine group or an almost simple classical group. Moreover, when G is the smallest Ree group, D is isomorphic either to the 2-(6 2 , 6, 2) design or to one of the three 2-(6 2 , 6, 6) designs constructed in this paper. All the four 2-designs have the 36 secants of a nondegenerate conic C of P G 2 (8) as a point set and 6-sets of secants in a remarkable configuration as a block set.