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AN APPROACH TO CAPABLE GROUPS AND SCHUR’S THEOREM
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 92 / Issue 1 / August 2015
- Published online by Cambridge University Press:
- 05 May 2015, pp. 52-56
- Print publication:
- August 2015
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- Article
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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 
$n\in \mathbf{N}$.
