Hall classes of groups with a locally finite obstruction (Q6622303)

This paper explores Hall classes in group theory, particularly focusing on extensions involving locally finite groups. A group class \(\mathfrak{X}\) is called a Hall class if it contains every group \(G\) having a nilpotent normal subgroup \(N\) such that \(G/N^{\prime}\) belongs to \(\mathfrak{X}\).\N\NThere are numerous examples showing that finite-by-\(\mathfrak{X}\) groups (i.e., extensions of finite groups by groups from class \(\mathfrak{X}\)) do not form a Hall class for many natural Hall classes \(\mathfrak{X}\).\N\NThe main result of the paper shows that if finite groups are replaced by locally finite groups or locally finite groups of finite exponent, the situation changes significantly. Specifically, the class of all (locally finite)-by-\(\mathfrak{X}\) groups and the class of all (locally finite of finite exponent)-by-\(\mathfrak{X}\) groups are Hall classes for the following \(18\) choices for \(\mathfrak{X}\): nilpotent, Fitting, hypercentral, Engel, Baer, Gruenberg, locally nilpotent, paranilpotent, hypercyclic, locally supersoluble, FC-nilpotent, FC-hypercentral, locally (nilpotent-by-finite), soluble, hyperabelian, locally soluble, finite (Prüfer) rank, finite abelian section rank.