The structure of groups with all proper quotients virtually nilpotent

Abstract

Just infinite groups play a significant role in profinite group theory. For each $c \geq 0$, we consider more generally JNN$_c$F 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\nbd$c$ nilpotent if and only if there are only finitely many subgroups arising as the lower central series terms $\gamma_{c+1}(K)$ of open normal subgroups $K$ of $G$. Based on this we prove several structure theorems. For instance, we characterize the JNN$_c$F profinite groups in terms of subgroups of the above form $\gamma_{c+1}(K)$. We also give a description of JNN$_c$F profinite groups as suitable inverse limits of virtually nilpotent profinite groups. Analogous results are established for the family of hereditarily JNN$_c$F groups and, for instance, we show that a Fitting-free JNN$_c$F profinite (or discrete) group is hereditarily JNN$_cF$ if and only if every maximal subgroup of finite index is JNN$_c$F. Finally, we give a construction of hereditarily JNN$_c$F groups, which uses as an input known families of hereditarily just infinite groups.