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

TL;DR

This paper classifies flag-transitive 2-designs with parameters where λ divides k, showing they are mostly affine or classical groups, with special cases involving the smallest Ree group and specific geometric configurations.

Contribution

It provides a complete classification of such 2-designs, identifying new examples related to the smallest Ree group and geometric configurations in projective space.

Findings

Most designs have affine or classical automorphism groups.

Special cases involve the smallest Ree group with unique geometric structures.

Identifies four specific 2-designs with geometric point sets and secant configurations.

Abstract

It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group $G$ of a $2$-$(k^{2}, k, \lambda)$ design D, with $\lambda \mid k$, is either an affine group or an almost simple classical group. Moreover, when $G$ is the smallest Ree group, $\mathcal{D}$ is isomorphic either to the $2$-$(62, 6, 2)$ design or to one of the three $2$- $(62, 6, 6)$ designs constructed in this paper. All the four $2$-designs have the $36$ secants of a nondegenerate conic $\mathcal{C}$ of $PG_{2}(8)$ as a point set and 6-sets of secants in a remarkable configuration as a block set.