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

TL;DR

This paper classifies non-trivial 2-(k^2, k, λ) designs with λ dividing k that admit a flag-transitive almost simple automorphism group, providing a comprehensive understanding of their structure.

Contribution

The paper offers a complete classification of these specific designs, expanding the knowledge of symmetric combinatorial structures with automorphism groups.

Findings

Complete classification of the specified designs.

Identification of conditions for flag-transitivity.

Insights into automorphism group structures.

Abstract

Non-trivial $2$-$(k^{2},k,\lambda )$ designs, with $\lambda \mid k$, admitting a flag-transitive almost simple automorphism group are classified.