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

 

Derived subgroups and centers of capable groups


Author: I. M. Isaacs
Journal: Proc. Amer. Math. Soc. 129 (2001), 2853-2859
MSC (2000): Primary 20D99
Published electronically: February 22, 2001
MathSciNet review: 1840087
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Abstract: A group $G$ is said to be capable if it is isomorphic to the central factor group $H/\mathbf{Z}(H)$ for some group $H$. It is shown in this paper that if $G$ is finite and capable, then the index of the center $\mathbf{Z}(G)$in $G$ is bounded above by some function of the order of the derived subgroup $G'$. If $G'$ is cyclic and its elements of order $4$ are central, then, in fact, $\vert G:\mathbf{Z}(G)\vert \le \vert G'\vert^{2}$.


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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/S0002-9939-01-05888-9
PII: S 0002-9939(01)05888-9
Received by editor(s): December 20, 1999
Received by editor(s) in revised form: February 22, 2000
Published electronically: February 22, 2001
Additional Notes: This paper was written with the partial support of the U.S. National Security Agency.
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2001 American Mathematical Society