The nilpotency of some groups with all subgroups subnormal

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 ¯nite 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

Abnormal subgroups and Carter subgroups in some infinite groups

Some properties of abnormal subgroups in generalized soluble groups will be considered. In particular, the transitivity of abnormality in metahypercentral groups is proven. Also it will be proven that a subgroup H of a radical group G is abnormal in G if and only if every intermediate subgroup for H coincides with its normalizer in G. This result will extend on radical groups the well-known criterion of abnormality for finite soluble groups obtained by D. Taunt. For some infinite groups (not only periodic) the existence of Carter subgroups and their conjugations will be also proven

On Groups whose Contranormal Subgroups are Normally Complemented

2000 Mathematics Subject Classification: 20F16, 20E15.Groups in which every contranormal subgroup is normally complemented has been considered. The description of such groups G with the condition Max-n and such groups having an abelian nilpotent residual satisfying Min-G have been obtained

Linear groups with the maximal condition on subgroups of infinite central dimension

Let A a vector space over a field F and let H be a subgroup of GL(F, A). We define centdimF H to be dimF (A/CA (H)). We say that H has finite central dimension if centdimF H is finite and we say that H has infinite central dimension otherwise. We consider soluble linear groups, in which the (ordered by inclusion) set of all subgroups having infinite central dimension satisfies the maximal condition

On the relationships between the factors of upper and lower central series in groups and other algebraic structures

We discuss some new recent development of the well known classical R. Baer and B. Neumann theorems. More precisely, we want to show the thematic which evolved from the mentioned classical results describing the relations between the central factor-group and the derived subgroup in an in nite group. We track these topics not only in groups, but also in some other algebraic structure

The influence of arrangement of subgroups on the group structure

Investigation of groups satisfying certain related to arrangement of subgroups conditions allows algebraists to introduce and describe many important classes of groups. Most of these conditions are based on the fundamental notion of normality and built with the help of this concept di erent subgroup chains (series). Some of important results obtained on this way we will discuss in the current surve

Direct decompositions of artinian modules related to formations of groups

We survey direct decompositions of artinian modules over group rings into two summands where all the chief factors of the first are X–central and all the chief factors of the other is X–eccentric, where X is a certain formation of finite groups

The nilpotency of some groups with all subgroups subnormal

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 ¯nite 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

Groups with a maximality condition for some non-normal subgroups

We describe (generalized) soluble-by-finite groups in which the set of non-normal subgroups which are not finitely generated satisfies the maximal condition