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1
Content available remote

The abelianization of hypercyclic groups

100%
Wehrfritz B.
Open Mathematics
|
2007
|
tom 5
|
nr 4
686-695
EN
Let G be a hypercyclic group. The most substantial results of this paper are the following. a) If G/G′ is 2-divisible, then G is 2-divisible. b) If G/G′ is a 2′-group, then G is a 2′-group. c) If G/G′ is divisible by finite-of-odd-order, then G/V is divisible by finite-of-odd-order, where V is the intersection of the lower central series (continued transfinitely) of O 2′ (G).
2
Content available remote

Groups with all subgroups permutable or of finite rank

81%
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.
3
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Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank)

81%
Badis A., Trabelsi N.
Open Mathematics
|
2011
|
tom 9
|
nr 6
1344-1348
EN
Our main result is that a locally graded group whose proper subgroups are Baer-by-Chernikov is itself Baer-by-Chernikov. We prove also that a locally (soluble-by-finite) group whose proper subgroups are Baer-by-(finite rank) is itself Baer-by-(finite rank) if either it is locally of finite rank but not locally finite or it has no infinite simple images.
4
Content available remote

The divisible radical of a group

81%
Wehrfritz B. A. F.
Open Mathematics
|
2009
|
tom 7
|
nr 3
387-394
EN
We consider the existence or otherwise of canonical divisible normal subgroups of groups in general. We present more counterexamples than positive results. These counterexamples constitute the substantive part of this paper.
5
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On locally finite minimal non-solvable groups

62%
Arıkan A., Sezer S., Smith H.
Open Mathematics
|
2010
|
tom 8
|
nr 2
266-273
EN
In the present work we consider infinite locally finite minimal non-solvable groups, and give certain characterizations. We also define generalizations of the centralizer to establish a result relevant to infinite locally finite minimal non-solvable groups.
6
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Groups with every subgroup ascendant-by-finite

62%
Camp-Mora S.
Open Mathematics
|
2013
|
tom 11
|
nr 12
2182-2185
EN
A subgroup H of a group G is called ascendant-by-finite in G if there exists a subgroup K of H such that K is ascendant in G and the index of K in H is finite. It is proved that a locally finite group with every subgroup ascendant-by-finite is locally nilpotent-by-finite. As a consequence, it is shown that the Gruenberg radical has finite index in the whole group.
7
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Groups with small deviation for non-subnormal subgroups

62%
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.
8
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Locally finite groups with all subgroups either subnormal or nilpotent-by-Chernikov

62%
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.
9
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The rank of the factor-group modulo the hypercenter and the rank of the some hypocenter of a group

62%
Kurdachenko L., Otal J.
Open Mathematics
|
2013
|
tom 11
|
nr 10
1732-1741
EN
We compare the special rank of the factors of the upper central series and terms of the lower central series of a group. As a consequence we are able to show some generalizations of a theorem of Reinhold Baer.
10
Content available remote

On some properties of pronormal subgroups

52%
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.
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