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1
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Groups with all subgroups permutable or of finite rank

100%
Dixon M., Karatas Y.
Open Mathematics
|
2012
|
tom 10
|
nr 3
950-957
EN
In this paper we investigate the structure of X-groups in which every subgroup is permutable or of finite rank. We show that every subgroup of such a group is permutable.
2
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Groups whose all subgroups are ascendant or self-normalizing

100%
Kurdachenko L., Otal J., Russo A., Vincenzi G.
Open Mathematics
|
2011
|
tom 9
|
nr 2
420-432
EN
This paper studies groups G whose all subgroups are either ascendant or self-normalizing. We characterize the structure of such G in case they are locally finite. If G is a hyperabelian group and has the property, we show that every subgroup of G is in fact ascendant provided G is locally nilpotent or non-periodic. We also restrict our study replacing ascendant subgroups by permutable subgroups, which of course are ascendant [Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972, 125(1), 1–16].
3
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On the structure of groups whose non-abelian subgroups are subnormal

100%
Kurdachenko L., Atlıhan S., Semko N.
Open Mathematics
|
2014
|
tom 12
|
nr 12
1762-1771
EN
The main aim of this article is to examine infinite groups whose non-abelian subgroups are subnormal. In this sense we obtain here description of such locally finite groups and, as a consequence we show several results related to such groups.
4
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Groups with small deviation for non-subnormal subgroups

76%
Kurdachenko L., Smith H.
Open Mathematics
|
2009
|
tom 7
|
nr 2
186-199
EN
We introduce the notion of the non-subnormal deviation of a group G. If the deviation is 0 then G satisfies the minimal condition for nonsubnormal subgroups, while if the deviation is at most 1 then G satisfies the so-called weak minimal condition for such subgroups (though the converse does not hold). Here we present some results on groups G that are either soluble or locally nilpotent and that have deviation at most 1. For example, a torsion-free locally nilpotent with deviation at most 1 is nilpotent, while a Baer group with deviation at most 1 has all of its subgroups subnormal.
5
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Locally finite groups with all subgroups either subnormal or nilpotent-by-Chernikov

76%
Cutolo G., Smith H.
Open Mathematics
|
2012
|
tom 10
|
nr 3
942-949
EN
Let G be a locally finite group satisfying the condition given in the title and suppose that G is not nilpotent-by-Chernikov. It is shown that G has a section S that is not nilpotent-by-Chernikov, where S is either a p-group or a semi-direct product of the additive group A of a locally finite field F by a subgroup K of the multiplicative group of F, where K acts by multiplication on A and generates F as a ring. Non-(nilpotent-by-Chernikov) extensions of this latter kind exist and are described in detail.
6
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On some properties of pronormal subgroups

64%
Kurdachenko L., Pypka A., Subbotin I.
Open Mathematics
|
2010
|
tom 8
|
nr 5
840-845
EN
New results on tight connections among pronormal, abnormal and contranormal subgroups of a group have been established. In particular, new characteristics of pronormal and abnormal subgroups have been obtained.
7
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Maximal subgroups and PST-groups

64%
Ballester-Bolinches A., Beidleman J., Esteban-Romero R., Pérez-Calabuig V.
Open Mathematics
|
2013
|
tom 11
|
nr 6
1078-1082
EN
A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19–25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versions of Kaplan’s results, which enables a better understanding of the relationships between these classes.
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