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The depth of a finite simple group

Authors: Timothy C. Burness, Martin W. Liebeck and Aner Shalev
Journal: Proc. Amer. Math. Soc. 146 (2018), 2343-2358
MSC (2010): Primary 20E32, 20E15; Secondary 20E28
Published electronically: February 16, 2018
MathSciNet review: 3778139
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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.

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Additional Information

Timothy C. Burness
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

Martin W. Liebeck
Affiliation: Department of Mathematics, Imperial College, London SW7 2BZ, United Kingdom

Aner Shalev
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel

Received by editor(s): August 2, 2017
Received by editor(s) in revised form: August 21, 2017
Published electronically: February 16, 2018
Additional Notes: The first and third authors acknowledge the hospitality and support of Imperial College, London, while part of this work was carried out. The third author acknowledges the support of ISF grant 1117/13 and the Vinik chair of mathematics which he holds.
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2018 American Mathematical Society