On a class of generalized nilpotent groups
The authors work in the class \(A\) (just my notation for this review only) of locally finite groups that are radical (that is, are hyper locally nilpotent) and which satisfy the minimal condition on \(p\)-subgroups for every prime \(p\). Within this class they consider the class \(B\) of all groups \(G\) with the property that every proper subgroup lies in a proper normal subgroup of \(G\). Note that \(A\) is a class of generalized soluble groups and \(B\) is a class of generalized nilpotent groups. The authors' aim is to show that within \(A\) the class \(B\) behaves very much as the class of finite nilpotent groups does within the class of finite soluble groups. In a whole host of ways I think the authors fully succeed with their objective. Of course minor modifications have to be made here and there to allow for the lack of finiteness. Perhaps the most dramatic of these is the replacement of the standard Frattini subgroup by the Frattini-like subgroup introduced by \textit{M. J. Tomkinson} [see Math. Proc. Camb. Philos. Soc 77, 247-257 (1975; Zbl 0301.20018)].
scientific article; zbMATH DE number 1763583