The depth of a finite simple group
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- by Timothy C. Burness, Martin W. Liebeck and Aner Shalev PDF
- Proc. Amer. Math. Soc. 146 (2018), 2343-2358 Request permission
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.References
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Additional Information
- Timothy C. Burness
- Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
- MR Author ID: 717073
- Email: t.burness@bristol.ac.uk
- Martin W. Liebeck
- Affiliation: Department of Mathematics, Imperial College, London SW7 2BZ, United Kingdom
- MR Author ID: 113845
- ORCID: 0000-0002-3284-9899
- Email: m.liebeck@imperial.ac.uk
- Aner Shalev
- Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
- MR Author ID: 228986
- ORCID: 0000-0001-9428-2958
- Email: shalev@math.huji.ac.il
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 2343-2358
- MSC (2010): Primary 20E32, 20E15; Secondary 20E28
- DOI: https://doi.org/10.1090/proc/13937
- MathSciNet review: 3778139