### On minimal Artinian modules and minimal Artinian linear groups.

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2010 Mathematics Subject Classification: Primary 20N25; Secondary 08A72, 03E72.

This article is dedicated to some criteria of generalized nilpotency involving pronormality and abnormality. Also new results on groups, in which abnormality is a transitive relation, have been obtained.

This article is dedicated to soluble groups, in which pronormality is a transitive relation. Complete description of such groups is obtained.

The article is dedicated to groups in which the set of abnormal and normal subgroups ($U$-subgroups) forms a lattice. A complete description of these groups under the additional restriction that every counternormal subgroup is abnormal is obtained.

A modular analogue of the well-known group theoretical result about finiteness of the derived subgroup in a group with a finite factor by its center has been obtained.

In this paper we obtain the description of the Leibniz algebras whose subalgebras are ideals.

This article discusses the Leibniz algebras whose upper hypercenter has finite codimension. It is proved that such an algebra $L$ includes a finite dimensional ideal $K$ such that the factor-algebra $L/K$ is hypercentral. This result is an extension to the Leibniz algebra of the corresponding result obtained earlier for Lie algebras. It is also analogous to the corresponding results obtained for groups and modules.

We begin to study the structure of Leibniz algebras having maximal cyclic subalgebras.

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