65 research outputs found
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
On non-periodic groups whose finitely generated subgroups are either permutable or pronormal
summary:The current article considers some infinite groups whose finitely generated subgroups are either permutable or pronormal. A group is called a generalized radical, if has an ascending series whose factors are locally nilpotent or locally finite. The class of locally generalized radical groups is quite wide. For instance, it includes all locally finite, locally soluble, and almost locally soluble groups. The main result of this paper is the following\endgraf Theorem. Let be a locally generalized radical group whose finitely generated subgroups are either pronormal or permutable. If is non-periodic then every subgroup of is permutable
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