## Abstract

Just infinite groups play a significant role in profinite group theory. For each c ≥ 0, we consider more generally JNN

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

_{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-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_{c}F profinite groups in terms of subgroups of the above form_{γc+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.Original language | English |
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Pages (from-to) | 147-189 |

Number of pages | 43 |

Journal | Pacific Journal of Mathematics |

Volume | 325 |

Issue number | 1 |

DOIs | |

Publication status | Published - 3 Sept 2023 |

## Keywords

- Profinite groups
- Residually finite groups
- Just infinite groups
- Just non-nilpotent-by-finite groups
- Virtually nilpotent groups
- Inverse system characterizations