A generalized Noetherian condition for Lie algebras

A Lie algebra (over any field and of any dimension) is Noetherian if it satisfies the maximal condition on ideals. We introduce a new and more general class of quasi-Noetherian Lie algebras that possess several of the main properties of Noetherian Lie algebras. This class is shown to be closed under quotients and extensions. We obtain conditions under which a quasi-Noetherian Lie algebra is Noetherian. Next, we consider various questions about locally nilpotent and soluble radicals of quasi-Noetherian Lie algebras. We show that there exists a semisimple quasi-Noetherian Lie algebra that is not Noetherian. Finally, we consider some analogous results for groups and prove that a quasi-Noetherian group is countably recognizable.

Groups Satisfying the Maximal Condition on Non-modular Subgroups

In this paper, (generalized) soluble groups for which the set of non-modular subgroups verifies the maximal condition and groups for which the set of non-permutable subgroups satisfies the same property are classified.

GROUPS WITH THE MAXIMAL CONDITION ON NON-BFC SUBGROUPS. II

A group $G$ is called a group with boundedly finite conjugacy classes (or a BFC-group) if $G$ is finite-by-abelian. A group $G$ satisfies the maximal condition on non-BFC-subgroups if every ascending chain of non-BFC-subgroups terminates in finitely many steps. In this paper the authors obtain the structure of finitely generated soluble-by-finite groups with the maximal condition on non-BFC subgroups.AMS 2000 Mathematics subject classification: Primary 20E15. Secondary 20F16; 20F24

A solution of a problem of Plotkin and Vovsi and an application to varieties of groups

Let K be an arbitrary field of characteristic 2, F a free group of countably infinite rank. We construct a finitely generated fully invariant subgroup U in F such that the relatively free group F/U satisfies the maximal condition on fully invariant subgroups but the group algebra K (F/U) does not satisfy the maximal condition on fully invariant ideals. This solves a problem posed by Plotkin and Vovsi. Using the developed techniques we also construct the first example of a non-finitely based (nilpotent of class 2)-by-(nilpotent of class 2) variety whose Abelian-by-(nilpotent of class at most 2) groups form a hereditarily finitely based subvariety.