Automorphism subgroups for designs with $\lambda=1$

TL;DR

This paper investigates the automorphism groups of certain combinatorial designs, proving that for large enough parameters, these designs can have automorphism groups containing specified subgroups, depending on the group's order.

Contribution

It establishes the existence of 2-(v,k,1) designs with automorphism groups containing a given subgroup, extending previous results to larger classes of groups.

Findings

Existence of designs with automorphism groups containing a specified subgroup for large v.

Automorphism groups can include subgroups isomorphic to any odd order group under certain conditions.

Weaker results are obtained for even order groups with additional restrictions.

Abstract

Given an integer $k\ge3$ and a group $G$ of odd order, if there exists a $2$-$(v,k,1)$-design and if $v$ is sufficiently large, then there is such a design whose automorphism group has a subgroup isomorphic to $G$. A weaker result is proved when $|G|$ is even and $(k,|G|)=1$.