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

 

Subgroups generated by small classes in finite groups


Author: I. M. Isaacs
Journal: Proc. Amer. Math. Soc. 136 (2008), 2299-2301
MSC (2000): Primary 20D25
Published electronically: March 14, 2008
MathSciNet review: 2390495
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Abstract: Let $ M(G)$ be the subgroup of $ G$ generated by all elements that lie in conjugacy classes of the two smallest sizes. Avinoam Mann showed that if $ G$ is nilpotent, then $ M(G)$ has nilpotence class at most $ 3$. Using a slight variation on Mann's methods, we obtain results that do not require us to assume that $ G$ is nilpotent. We show that if $ G$ is supersolvable, then $ M(G)$ is nilpotent with class at most $ 3$, and in general, the Fitting subgroup of $ M(G)$ has class at most $ 4$.


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Additional Information

I. M. Isaacs
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: isaacs@math.wisc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09263-0
PII: S 0002-9939(08)09263-0
Keywords: Conjugacy class size, nilpotent, supersolvable, nilpotence class
Received by editor(s): March 26, 2007
Published electronically: March 14, 2008
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2008 American Mathematical Society