MaxDim of some simple groups

TL;DR

This paper investigates the MaxDim invariant for certain simple groups, providing lower bounds, detailed computations, and estimations for groups like the Mathieu 24, McLaughlin, and Higman-Sims, using their actions on combinatorial structures.

Contribution

It introduces new bounds and computations for MaxDim of sporadic groups, especially leveraging their actions on combinatorial objects like graphs and Steiner systems.

Findings

Lower bounds for MaxDim in Suzuki tower groups

Exact MaxDim for Mathieu 24 group

Estimations for McLaughlin and Higman-Sims groups

Abstract

MaxDim, along with it relatives m and i, are group invariants analogous to the concept of dimension in vector spaces. In this paper we give the following result on the MaxDim of sporadic groups: 1. An inductive argument for the lower bounds of the MaxDim of those groups in the Suzuki tower, utilizing the fact that those groups act on strongly regular graphs. 2. A detailed computation of the MaxDim of Mathieu 24 group, utilizing the fact that it acts as automorphisms of Steiner system. 3. Estimations of the MaxDim for McLaughlin and Higmann-Sims Group.