Flag-transitive point-primitive non-symmetric 2-(v,k,2) designs with alternating socle

TL;DR

This paper classifies certain flag-transitive point-primitive non-symmetric 2-(v,k,2) designs with alternating socle, showing they are limited to specific small parameters.

Contribution

It proves that such designs with alternating socle are only possible for parameters 2-(6,3,2) or 2-(10,4,2), providing a classification result.

Findings

Designs are limited to 2-(6,3,2) and 2-(10,4,2) cases.

Automorphism groups have alternating socle A_n with n≥5.

Non-symmetric flag-transitive point-primitive designs are highly restricted.

Abstract

This paper studies flag-transitive point-primitive non-symmetric $2$-($v,k,2$) designs. We prove that if $\mathcal{D}$ is a non-trivial non-symmetric $2$-$(v,k,2)$ design admitting a flag-transitive point-primitive automorphism group $G$ with $Soc(G)=A_{n}$ for $n\geq5$, then $\mathcal{D}$ is a $2$-$(6,3,2)$ or $2$-$(10,4,2)$ design.