Automorphism groups of designs with $\lambda=1$

TL;DR

This paper demonstrates that for certain parameters, any finite group can be realized as the automorphism group of an infinite family of 2-designs with specific parameters, expanding understanding of symmetry groups in combinatorial designs.

Contribution

It constructs infinite families of 2-(v,k,1) designs with prescribed automorphism groups for particular parameters, showing the universality of automorphism groups in these designs.

Findings

Any finite group can be realized as an automorphism group for infinitely many 2-(v,k,1) designs with specified parameters.

The construction applies when k=q>2 or k=q+1 for a prime power q.

The result broadens the class of known automorphism groups for combinatorial designs.

Abstract

If $G$ is a finite group and $k =q>2$ or $k=q+1$ for a prime power $q$ then, for infinitely many integers $v$, there is a $2$-$(v,k,1)$-design ${\bf D}$ for which ${\rm Aut} {\bf D}\cong G$.