Relationships between the factors of the upper and the lower central series of a group
A celebrated theorem, essentially due to \textit{I. Schur} [J. Reine Angew. Math. 127, 20--50 (1904; JFM 35.0155.01)], shows that if \(G\) is any group whose centre \(Z(G)\) has finite index, then the commutator subgroup \(G'\) of \(G\) is finite. This result was later extended by \textit{R. Baer} [Math. Ann. 124, 161--177 (1952; Zbl 0046.02202)], who proved that if the term \(Z_k(G)\) of the upper central series of the group \(G\) has finite index, then the term \(\gamma_{k+1}(G)\) of the lower central series of \(G\) is finite. Inspired by Schur's theorem, \textit{S. Franciosi} et al. [J. Algebra 174, No. 3, 823--847 (1995; Zbl 0831.20043)] called a group class \(\mathfrak X\) a Schur class if whenever the factor group \(G/Z(G)\) belongs to \(\mathfrak X\), also \(G'\) is an \(\mathfrak X\)-group. In the paper under review, a group class \(\mathfrak X\) is called a Baer class if for each group \(G\) such that \(G/Z_k(G)\) is an \(\mathfrak X\)-group for some positive integer \(k\), also the subgroup \(\gamma_{k+1}(G)\) belongs to \(\mathfrak X\). Clearly, all Baer classes have the Schur property, and it is a very interesting problem to determine which Schur classes are also Baer classes. It was proved by \textit{A. Mann} [J. Group Theory 10, No. 4, 435--436 (2007; Zbl 1160.20030)] that locally finite groups of finite exponent form a Schur class, and the authors prove here that such class has also the Baer property.
scientific article; zbMATH DE number 6625455
scientific article