Schur's theorem and Wiegold's bound
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 \(| G' |\leq t^{(t_{\ast}-1)/2}\).
scientific article; zbMATH DE number 6893119