The depth of a finite simple group

TL;DR

This paper introduces the concept of group depth for finite groups, investigates its properties for simple groups, and establishes bounds, revealing new insights into the structure of these groups.

Contribution

It defines the depth of finite groups, characterizes minimal depth simple groups, and provides bounds for the depth of groups of Lie type, connecting group theory with number theory.

Findings

Alternating groups have bounded depth.

Bounds are established for groups of Lie type.

The depth relates to the length of simple groups.

Abstract

We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups. We determine the simple groups of minimal depth, and show, somewhat surprisingly, that alternating groups have bounded depth. We also establish general upper bounds on the depth of simple groups of Lie type, and study the relation between the depth and the much studied notion of the length of simple groups. The proofs of our main theorems depend (among other tools) on a deep number-theoretic result, namely, Helfgott's recent solution of the ternary Goldbach conjecture.