Groups of class 2n in which all proper subgroups have class at most n

摘要

Let n = 3 be a positive integer. We show that there exist nilpotent groups of class 2n in which every proper subgroup has class at most n. (Such groups are necessarily finite.) We also show that there exists a torsion-free nilpotent group of class 2n in which every subgroup of infinite index has class at most n. These results are proved by establishing analogous result for nilpotent Lie algebras and then using the Lazard correspondence or the Mal'cev correspondence as appropriate. (C) 2017 Elsevier Inc. All rights reserved.