What question did this study set out to answer?

The aim is to examine the depth of core-free subgroups within simple groups and derive new results.

February 22, 2026Open Access

On the depth of subgroups of simple groups

Key Points

The aim is to examine the depth of core-free subgroups within simple groups and derive new results.
Applied computational techniques for subgroup depth analysis.
Used probabilistic methods initially designed for permutation groups.
Calculated exact depth for every subgroup of almost simple sporadic groups.
Established best bounds on the depth of irreducible subgroups of classical groups.
Calculated exact depths for subgroups of almost simple sporadic groups except a few cases.
Identified a range of new results on subgroup depths, significantly extending previous findings.

Abstract

Abstract The depth of a subgroup H of a finite group G is a positive integer defined with respect to the inclusion of the corresponding complex group algebras CH CG. This notion was originally introduced by Boltje, Danz and Külshammer in 2011, and it has been the subject of numerous papers in recent years. In this paper, we study the depth of core-free subgroups, which allows us to apply powerful computational and probabilistic techniques that were originally designed for studying bases for permutation groups. We use these methods to prove a wide range of new results on the depth of subgroups of almost simple groups, significantly extending the scope of earlier work in this direction. For example, we establish best possible bounds on the depth of irreducible subgroups of classical groups and primitive subgroups of symmetric groups. And with the exception of a handful of open cases involving the Baby Monster, we calculate the exact depth of every subgroup of every almost simple sporadic group. We also present a number of open problems and conjectures.

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