The structure of groups with all proper quotients virtually nilpotent

Just infinite groups play a significant role in profinite group theory. For each c ≥ 0, we consider more generally JNN_cF profinite (or, in places, discrete) groups that are Fitting-free; these are the groups G such that every proper quotient of G is virtually class-c nilpotent whereas G itself is not, and additionally G does not have any non-trivial abelian normal subgroup. When c = 1, we obtain the just non-(virtually abelian) groups without non-trivial abelian normal subgroups.

Our first result is that a finitely generated profinite group is virtually class-c nilpotent if and only if there are only finitely many subgroups arising as the lower central series terms _γc+1(K) of open normal subgroups K of G. Based on this we prove several structure theorems. For instance, we characterize the JNN_cF profinite groups in terms of subgroups of the above form _γc+1(K). We also give a description of JNN_cF profinite groups as suitable inverse limits of virtually nilpotent profinite groups. Analogous results are established for the family of hereditarily JNN_cF groups and, for instance, we show that a Fitting-free JNN_cF profinite (or discrete) group is hereditarily JNN_cF if and only if every maximal subgroup of finite index is JNN_cF. Finally, we give a construction of hereditarily JNN_cF groups, which uses as an input known families of hereditarily just infinite groups.