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

 

Bounds for the index of the centre in capable groups


Authors: K. Podoski and B. Szegedy
Journal: Proc. Amer. Math. Soc. 133 (2005), 3441-3445
MSC (2000): Primary 20E34, 20D60, 20D15, 20D25
Published electronically: July 13, 2005
MathSciNet review: 2163577
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Abstract: A group $H$ is called capable if it is isomorphic to $G/\mathbf{Z}(G)$for some group $G$. Let $H$ be a capable group. I. M. Isaacs (2001) showed that if $H$ is finite, then the index of the centre is bounded above by some function of $\vert H'\vert$. We show that if $\vert H'\vert<\infty$, then $\vert H:Z(H)\vert\leq \vert H'\vert^{c\log_2\vert H'\vert}$ with some constant $c$ and this bound is essentially best possible. We complete a result of Isaacs, showing that if $H'$ is a cyclic group, then $\vert H:\mathbf{Z}(H)\vert\leq \vert H'\vert^2$.


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Additional Information

K. Podoski
Affiliation: Department of Algebra and Number Theory, Eötvös University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary
Address at time of publication: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, H-1364 Budapest, Hungary
Email: pcharles@cs.elte.hu, pcharles@renyi.hu

B. Szegedy
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, H-1364 Budapest, Hungary
Email: szegedy@renyi.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07663-X
PII: S 0002-9939(05)07663-X
Received by editor(s): March 10, 2003
Received by editor(s) in revised form: January 6, 2004
Published electronically: July 13, 2005
Additional Notes: This research was partially supported by the Hungarian National Research Foundation (OTKA), grant no. T038059
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2005 American Mathematical Society