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The depth of a finite simple group

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Original languageEnglish
Pages (from-to)2343-2358
Number of pages16
JournalProceedings of the American Mathematical Society
Issue number6
Early online date16 Feb 2018
DateAccepted/In press - 28 Aug 2017
DateE-pub ahead of print - 16 Feb 2018
DatePublished (current) - Jun 2018


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.

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