The structure of groups with all proper quotients virtually nilpotent

Benjamin Klopsch and Martyn Quick

Vol. 325 (2023), No. 1, 147–189

DOI: 10.2140/pjm.2023.325.147

Abstract

Just infinite groups play a significant role in profinite group theory. For each $c\ge 0$, 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 nontrivial abelian normal subgroup. When $c=1$, we obtain the just non-(virtually abelian) groups without nontrivial 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 ${\gamma }_{c+1}(K)$ 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${}_c$F 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.

Keywords

profinite groups, residually finite groups, just infinite groups, just non-nilpotent-by-finite groups, virtually nilpotent groups, inverse system characterizations

Mathematical Subject Classification

Primary: 20E18

Secondary: 20E26, 20F19

Milestones

Received: 15 December 2022

Revised: 1 August 2023

Accepted: 1 August 2023

Published: 3 September 2023