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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Powers in finite groups and a criterion for solubility


Authors: Martin W. Liebeck and Aner Shalev
Journal: Proc. Amer. Math. Soc. 141 (2013), 4179-4189
MSC (2010): Primary 20D10, 20E07, 20D06
Published electronically: September 10, 2013
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Abstract: We study the set $ G^{[k]}$ of $ k^{th}$ powers in finite groups $ G$. We prove that if $ G^{[12]}$ is a subgroup, then $ G$ must be soluble; moreover, 12 is the minimal number with this property. The proof relies on results of independent interest, classifying almost simple groups $ G$ and positive integers $ k$ for which $ G^{[k]}$ contains the socle of $ G$.


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Additional Information

Martin W. Liebeck
Affiliation: Department of Mathematics, Imperial College, London SW7 2AZ, 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: http://dx.doi.org/10.1090/S0002-9939-2013-11790-9
PII: S 0002-9939(2013)11790-9
Received by editor(s): February 12, 2012
Published electronically: September 10, 2013
Additional Notes: The authors are grateful for the support of an EPSRC grant. The second author acknowledges the support of grants from the Israel Science Foundation and ERC
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.