Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/proc/13937
Published electronically: February 16, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 20E32, 20E15, 20E28

Retrieve articles in all journals with MSC (2010): 20E32, 20E15, 20E28


Additional Information

Timothy C. Burness
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
Email: t.burness@bristol.ac.uk

Martin W. Liebeck
Affiliation: Department of Mathematics, Imperial College, London SW7 2BZ, United Kingdom
Email: m.liebeck@imperial.ac.uk

Aner Shalev
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
Email: shalev@math.huji.ac.il

DOI: https://doi.org/10.1090/proc/13937
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

American Mathematical Society