On the automorphisms of designs constructed from finite simple groups

TL;DR

This paper investigates the automorphism groups of combinatorial designs derived from finite simple groups, improving existing methods and applying them to recover specific Steiner systems associated with Mathieu groups.

Contribution

It introduces improved techniques for analyzing automorphism groups of designs from finite simple groups and applies these to identify certain Steiner systems linked to Mathieu groups.

Findings

Automorphism groups of designs from finite simple groups are characterized.

Enhanced method for studying automorphisms is developed.

Specific Steiner systems associated with Mathieu groups are retrieved.

Abstract

Here we study the automorphism groups of $1$-designs constructed from finite nonabelian simple groups by using two methods presented in Moori (Information Security, Coding Theory and Related Combinatorics, 2011). We obtain some general results for both and improve one of these methods. In an application to the sporadic Mathieu groups $M_{n}$, we are able to retrieve the Steiner systems $S(t,t+3,n)$ where $(n,t)\in\{(22,3),(23,4),(24,5)\}$.