Schur's theorem and Wiegold's bound
type: Item
instance of: scholarly article
title: Schur's theorem and Wiegold's bound
zbMATH Open document ID: 1392.20029
published in: Journal of Algebra
publication date: 2018-06-22T00:00:00Z
review text: Let \(G\) be a group and let \(Z(G)\) be the center of \(G\). If \(Z \leq Z(G)\) and \(| G : Z | =t < \infty\), then, by a famous theorem of Schur, the derived subgroup \(G'\) of \(G\) is finite of order bounded by a function of \(t\). The best known bound is \textit{J. Wiegold}'s bound [Math. Z. 89, 345--347 (1965; Zbl 0134.03002)], namely \(| G' | \leq t^{b}\), where \(b=(\log_{q} t - 1)/2\) and \(q\) denotes the least prime divisor of \(t\). In this nice paper, the author presents some improved bounds for \(| G'|\), together with short and elementary proofs of known bounds. Let \(\pi(t)\) be the set of primes dividing \(t\), \[ t=\prod_{p \in \pi(t)}p^{e(p)},\;\;\; e(t)=\max \{ e(p) \mid p \in \pi(t) \}, \;\;\; s=\Big (\sum_{p \in \pi(t)} e(p) \Big ) -e(t), \] \(q=\min \{ p \mid p \in \pi((t) \}\), \(t_{\ast}=\log_{q} t\) and \(\tau= \lfloor t_{\ast} \rfloor\). Then \(| G' |^{2}\) divides \(t^{e(t)+1}\); moreover if \(s\not =1\), then \(| G' |^{2}\) divides \(t^{\tau-1}\) and \(| G' |\leq t^{(\tau-1)/2}\), if \(s=1\), then \(| G' |^{2}\) divides \(t^{\tau}\) and \(| G' |\leq t^{(t_{\ast}-1)/2}\).
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 6893119
zbMATH Keywords: Central subgroup
zbMATH Keywords: derived subgroup
zbMATH Keywords: Shur's theorem
author: B. A. F. Wehrfritz
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2018.02.023
OpenAlex ID: W2792940823
cites work: Q5654235
cites work: Group and ring theoretic properties of polycyclic groups.
cites work: Multiplicators and groups with finite central factor-groups
Wikidata QID: Q130089038
DOI: 10.1016/J.JALGEBRA.2018.02.023
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description: scientific article; zbMATH DE number 6893119