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

MathSciNet review:
3105861

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Abstract: We study the set of powers in finite groups . We prove that if is a subgroup, then must be soluble; moreover, 12 is the minimal number with this property. The proof relies on results of independent interest, classifying almost simple groups and positive integers for which contains the socle of .

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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:
https://doi.org/10.1090/S0002-9939-2013-11790-9

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.