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AN APPROACH TO CAPABLE GROUPS AND SCHUR’S THEOREM

Part of: Structure and classification of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  05 May 2015

MITRA HASSANZADEH  and
RASOUL HATAMIAN
MITRA HASSANZADEH*
Affiliation:
Department of Pure Mathematics, Ferdowsi University of Mashhad, PO Box 1159-91775, Mashhad, Iran email mtr.hassanzadeh@gmail.com
RASOUL HATAMIAN
Affiliation:
Department of Pure Mathematics, Ferdowsi University of Mashhad, PO Box 1159-91775, Mashhad, Iran email hatamianr@yahoo.com
*
✉mtr.hassanzadeh@gmail.com
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Abstract

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Podoski and Szegedy [‘On finite groups whose derived subgroup has bounded rank’, Israel J. Math.178 (2010), 51–60] proved that for a finite group $G$ with rank $r$, the inequality $[G:Z_{2}(G)]\leq |G^{\prime }|^{2r}$ holds. In this paper we omit the finiteness condition on $G$ and show that groups with finite derived subgroup satisfy the same inequality. We also construct an $n$-capable group which is not $(n+1)$-capable for every $n\in \mathbf{N}$.

Keywords

derived subgroupSchur’s theoremn-isoclinismn-capable groups

MSC classification

Primary: 20D60: Arithmetic and combinatorial problems
Secondary: 20E34: General structure theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 1 , August 2015 , pp. 52 - 56
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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