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

 

Large subgroups of a finite group of even order


Authors: Bernhard Amberg and Lev Kazarin
Journal: Proc. Amer. Math. Soc. 140 (2012), 65-68
MSC (2010): Primary 20D05, 20D06
Published electronically: May 10, 2011
MathSciNet review: 2833517
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Abstract: It is shown that if $ G$ is a group of even order with trivial center such that $ \vert G\vert>2\vert C_{G}(t)\vert^{3}$ for some involution $ t\in G$, then there exists a proper subgroup $ H$ of $ G$ such that $ \vert G\vert< \vert H\vert^{2}$. If $ \vert G\vert>\vert C_{G}(t)\vert^{3}$ and $ k(G)$ is the class number of $ G$, then $ \vert G\vert\leq k(G)^{3}$.


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Additional Information

Bernhard Amberg
Affiliation: Fachbereich Mathematik, Universität Mainz, D-55099 Mainz, Germany
Email: amberg@mathematik.uni-mainz.de

Lev Kazarin
Affiliation: Department of Mathematics, Yaroslavl State University, 150000 Yaroslavl, Russia
Email: kazarin@uniyar.ac.ru

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10982-1
PII: S 0002-9939(2011)10982-1
Keywords: Group, subgroup, centralizer, involution, conjugacy class, square root
Received by editor(s): November 3, 2010
Published electronically: May 10, 2011
Additional Notes: The second author is grateful to the Department of Mathematics of the University of Mainz for its warm hospitality during the time when this reseach was done. He would also like to thank the Deutsche Forschungsgemeinschaft for its financial support.
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2011 American Mathematical Society