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Power sets and soluble subgroups


Authors: Martin W. Liebeck and Aner Shalev
Journal: Proc. Amer. Math. Soc. 142 (2014), 3757-3760
MSC (2010): Primary 20D10, 20E07, 20D06
DOI: https://doi.org/10.1090/S0002-9939-2014-12203-9
Published electronically: July 24, 2014
MathSciNet review: 3251717
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that for certain positive integers $ k$, such as 12, a normal subgroup of a finite group which consists of $ k^{th}$ powers is necessarily soluble. This gives rise to new solubility criteria, and solves an open problem from a 2013 paper by the authors.


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

  • [1] E. Hrushovski, P. H. Kropholler, A. Lubotzky, and A. Shalev, Powers in finitely generated groups, Trans. Amer. Math. Soc. 348 (1996), no. 1, 291-304. MR 1316851 (96f:20061), https://doi.org/10.1090/S0002-9947-96-01456-0
  • [2] M. W. Liebeck and A. Shalev, Powers in finite groups and a criterion for solubility, Proc. Amer. Math. Soc. 141 (2013), no. 12, 4179-4189. MR 3105861

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Additional Information

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

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

DOI: https://doi.org/10.1090/S0002-9939-2014-12203-9
Received by editor(s): December 20, 2012
Published electronically: July 24, 2014
Additional Notes: The authors are grateful for the support of an EPSRC grant
The second author acknowledges the support of Advanced ERC Grant 247034, an ISF grant 754/08, and the Miriam and Julius Vinik Chair in Mathematics, which he holds.
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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