Local nilpotent groups satisfying a weak condition of minimality or maximality for subgroups of fixed degree of nilpotency

Abstract

We study locally nilpotent groups containing subgroups of class $c, c>1$, and satisfying the weak maximum condition or the weak minimum condition on $с$-nilpotent subgroups. It is proved that nilpotent groups of this type are minimax and periodic locally nilpotent groups of this type are Chernikov groups. It is also proved that if a group $G$  is either nilpotent or periodic locally nilpotent and if all of its $с$-nilpotent subgroups are of finite rank, then $G$  is of finite rank. If $G$  is a non-periodic locally nilpotent group, these results, in general, are not valid.