On a generalization of Baer theorem. (Q2839343)

A relevant result of Reinhold Baer shows that if the term \(Z_n(G)\) of the upper central series of the group \(G\) has finite index in \(G\) for some positive integer \(n\), then the term \(\gamma_{n+1}(G)\) of the lower central series is finite, and so \(G\) is finite-by-nilpotent. This theorem has recently been generalized by \textit{M. De Falco, C. Musella, Y. P. Sysak} and the reviewer, who proved that if the hypercentre (i.e. the largest term of the upper central series) of the group \(G\) has finite index, then \(G\) contains a finite normal subgroup \(L\) such that \(G/L\) is hypercentral [Proc. Am. Math. Soc. 139, No. 2, 385-389 (2011; Zbl 1228.20023)].NEWLINENEWLINE In the paper under review, the authors provide a simpler proof of this latter result. Moreover, they also prove that in this situation, if the hypercentre has index \(t\), then the finite normal subgroup \(L\) can be chosen of order at most \(t^k\), where \(k=(\log_pt+1)/2\) and \(p\) is the least prime divisor of \(t\).