Bounds for the index of the centre in capable groups

Podoski, K.; Szegedy, B.

doi:10.1090/S0002-9939-05-07663-X

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 $|H:\mathbf {Z}(H)|\leq |H’|^2$.

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