Groups with prescribed quotient groups and associated module theory (Q2784583)

This monograph is devoted to the study of `just non-\(\mathfrak X\)-groups'. Here, \(\mathfrak X\) is a class of groups, and a group \(G\) is just non-\(\mathfrak X\) if \(G\not\in{\mathfrak X}\), but \(H\in{\mathfrak X}\) for every proper factor group \(H\) of \(G\). Of course, infinite simple groups are just non-\(\mathfrak X\) for every class \(\mathfrak X\), so in order to obtain worthwhile results it's necessary to make some hypothesis which ensures an adequate supply of proper factors. This is most commonly achieved by considering only groups \(G\) whose Fitting subgroup is non-trivial. The Fitting subgroup of \(G\) is generated by all the nilpotent normal subgroups of \(G\), so an equivalent hypothesis on \(G\) is the requirement that it contains a non-trivial Abelian normal subgroup \(A\).NEWLINENEWLINENEWLINESince \(A\) is then (via conjugation) a module for the group ring \(\mathbb{Z}(G/A)\) of the \(\mathfrak X\)-group \(G/A\), the first 2 of the 3 chapters of the book are concerned with modules over group rings of various classes of groups \(\mathfrak X\). This constitutes about half the length of the book. Various classes \(\mathfrak X\) are considered here, for example FC-hypercentral groups and polycyclic-by-finite groups amongst others. The motivation above leads the analysis to focus on two sorts of modules in particular -- simple modules, and just infinite modules, the latter being those which are infinite, have non-zero simple socle, and for which every proper submodule has finite index.NEWLINENEWLINENEWLINEIn the third chapter of the book this module theory is used to derive many results about just non-\(\mathfrak X\)-groups for numerous classes \(\mathfrak X\).