Classification of the non-trivial $2$-$(k^{2},k,\lambda )$ designs, with $\lambda \mid k$, admitting a flag-transitive automorphism group of affine type

TL;DR

This paper classifies certain symmetric combinatorial designs with specific automorphism groups, focusing on those with affine symmetry that are not contained in a known affine general linear group extension.

Contribution

It provides a classification of non-trivial 2-(k^2, k, λ) designs with λ dividing k, admitting a flag-transitive automorphism group of affine type outside a specific known subgroup.

Findings

Complete classification of the specified designs and automorphism groups.

Identification of conditions under which such designs exist.

Clarification of the automorphism group's structure in these cases.

Abstract

The pairs $(\mathcal{D},G)$, where $\mathcal{D}$ is a non-trivial $2$-$(k^{2},k,\lambda )$ design, with $\lambda \mid k$, and $G$ is a flag-transitive automorphism group of $\mathcal{D}$ of affine type such that $G \nleq A \Gamma L_{1}(k^{2})$, are classified.