Title: The depth of a finite simple group,
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.