The structure of groups with all proper quotients virtually nilpotent

Abstract

Just infinite groups play a significant role in profinite group theory. For each c ≥ 0, we consider more generally JNNcF 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 JNNcF profinite groups in terms of subgroups of the above form γc+1(K). We also give a description of JNNcF profinite groups as suitable inverse limits of virtually nilpotent profinite groups. Analogous results are established for the family of hereditarily JNNcF groups and, for instance, we show that a Fitting-free JNNcF profinite (or discrete) group is hereditarily JNNcF if and only if every maximal subgroup of finite index is JNNcF. Finally, we give a construction of hereditarily JNNcF groups, which uses as an input known families of hereditarily just infinite groups.