Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-05-07663-X
Published electronically: July 13, 2005
MathSciNet review: 2163577
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $\vert H:\mathbf{Z}(H)\vert\leq \vert H'\vert^2$.


References [Enhancements On Off] (What's this?)

  • 1. H. Heineken, Nilpotent groups of class two that can appear as central quotient groups, Rend. Sem. Mat. Univ. Padova 84 (1990), 241-248. MR 1101296 (92c:20068)
  • 2. I. M. Isaacs, Derived subgroups and centers of capable groups, Proc. Amer. Math. Soc. 129 (2001), 2853-2859. MR 1840087 (2002c:20035)
  • 3. I. D. Macdonald, Some explicit bounds in groups with finite derived groups, Proc. London Math. Soc (3) 11 (1961), 23-56. MR 0124433 (23:A1745)
  • 4. K. Podoski, Groups covered by an infinite number of Abelian subgroups, Combinatorica 21 (3) (2001), 413-416. MR 1848059 (2002e:20061)
  • 5. K. Podoski, B. Szegedy, Bounds in groups with finite Abelian coverings or with finite derived groups, J. Group Theory 5 (2002), 443-452. MR 1931369 (2003i:20052)
  • 6. Derek J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York (1982). MR 0648604 (84k:20001)
  • 7. J. Wiegold, Multiplicators and groups with finite central factor-groups, Math. Z. 89 (1965), 345-347. MR 0179262 (31:3510)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20E34, 20D60, 20D15, 20D25

Retrieve articles in all journals with MSC (2000): 20E34, 20D60, 20D15, 20D25


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: https://doi.org/10.1090/S0002-9939-05-07663-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

American Mathematical Society