Bounds for the index of the centre in capable groups
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- by K. Podoski and B. Szegedy PDF
- Proc. Amer. Math. Soc. 133 (2005), 3441-3445 Request permission
Abstract:
A group $H$ is called capable if it is isomorphic to $G/\mathbb {Z}$ 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 $|H’|$. We show that if $|H’|<\infty$, then $|H:Z(H)|\leq |H’|^{c\log _2|H’|}$ 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 $|H:\mathbf {Z}(H)|\leq |H’|^2$.References
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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
- 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
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 3441-3445
- MSC (2000): Primary 20E34, 20D60, 20D15, 20D25
- DOI: https://doi.org/10.1090/S0002-9939-05-07663-X
- MathSciNet review: 2163577