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

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

Let G be a group with all subgroups subnormal. A normal subgroup N of G is said to be G-minimax if it has a finite G-invariant series whose factors are abelian and satisfy either max-G or min- G. It is proved that if the normal closure of every element of G is G-minimax then G is nilpotent and the normal closure of every element is minimax. Further results of this type are also obtained.

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.

Let $G$ be a group with the property that there are no infinite descending chains of non-subnormal subgroups of $G$ for which all successive indices are infinite. The main result is that if $G$ is a locally (soluble-by-finite) group with this property then either $G$ has subgroups subnormal or $G$ is a soluble-by-finite minimax group. This result fills a gap left in an earlier paper by the same authors on groups with the stated property.

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 group $G$ has subnormal deviation at most $1$ if, for every descending chain ${H}_{0}>{H}_{1}>\cdots $ of non-subnormal subgroups of $G$, for all but finitely many $i$ there is no infinite descending chain of non-subnormal subgroups of $G$ that contain ${H}_{i+1}$ and are contained in ${H}_{i}$. This property $\U0001d513$, say, was investigated in a previous paper by the authors, where soluble groups with $\U0001d513$ and locally nilpotent groups with $\U0001d513$ were effectively classified. The present article affirms a conjecture from that article by showing that locally soluble-by-finite...

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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