On primary elements in groups

Shunkov, V. P.

doi:10.1007/bf01058709

Cited by 7 publications

(12 citation statements)

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“…In this article we generalize some results of [1][2][3][4][5][6][7][8][9][10] and prove the theorems that are announced in [11,12].…”

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confidence: 83%

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confidence: 92%

“…In an infinite group G take solvable elements a and b of prime orders so that |a|·|b| > 4 and for almost all c ∈ b G the subgroups a, c are finite. Then either G contains only finitely many finite order elements or at least one of a and b lies in an f -local subgroup that contains infinitely many finite order elements.A group G is said to possess periodic part whenever the set of finite order elements of G forms a subgroup [9]. Call a finite order nonidentity element a of some infinite group G generally finite if a lies in the base of a fan of finite subgroups whose amalgam almost completely contains a nonidentity conjugacy class of G. In other words, for almost all c in some nonidentity conjugacy class b G the subgroups a, c are finite; i.e., the (a, b)-finiteness condition [9] holds.…”

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confidence: 99%

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On the existence of f-local subgroups in a group

Sozutov¹,

Yanchenko²

2006

Sib Math J

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We prove the existence of infinite subgroups with nontrivial locally finite radicals and of locally finite subgroups in the groups with almost finite almost solvable elements of prime orders and in the groups with generally finite elements.In this article we generalize some results of [1-10] and prove the theorems that are announced in [11,12].If the set of finite order elements is finite in an infinite group G then by the well-known theorem of Dietzmann this set forms a finite fully characteristic subgroup of G. If the set of finite order elements in G is infinite then some questions concerning their location in the group are in order [9]. One of them is the well-known question of Kargapolov [13, Problem 1.24] about the existence of infinite abelian subgroups in an infinite group, which in the general case was answered negatively by Novikov and Adyan [14]. Therefore, it has to be considered separately for many classes of infinite groups. The question of Kargapolov is closely related to a weaker question of the existence of the f -local subgroups, i.e., the infinite subgroups with nontrivial locally finite radicals (see the question of Strunkov [13, Problem 2.75]). In this article we give positive answers to these questions in some classes of groups.Theorem 1. Given an infinite group G, take an element a of prime order p > 2 so that for almost all a g ∈ a G the subgroups a, a g are finite and solvable. Then either G contains only finitely many finite order elements or a lies in an f -local subgroup that contains infinitely many finite order elements.Call an element x of a group G solvable whenever all finite subgroups generated by elements in x G are solvable.Theorem 2. In an infinite group G take solvable elements a and b of prime orders so that |a|·|b| > 4 and for almost all c ∈ b G the subgroups a, c are finite. Then either G contains only finitely many finite order elements or at least one of a and b lies in an f -local subgroup that contains infinitely many finite order elements.A group G is said to possess periodic part whenever the set of finite order elements of G forms a subgroup [9]. Call a finite order nonidentity element a of some infinite group G generally finite if a lies in the base of a fan of finite subgroups whose amalgam almost completely contains a nonidentity conjugacy class of G. In other words, for almost all c in some nonidentity conjugacy class b G the subgroups a, c are finite; i.e., the (a, b)-finiteness condition [9] holds.Theorem 3. If all finite subgroups of G are solvable and each quotient of G by a finite subgroup either is torsion free or contains a generally finite element of odd prime order then G either possesses finite periodic part or includes an infinite locally finite subgroup.In the case that the general finiteness holds for all prime order elements in G and is inherited by all subgroups of G and all its quotients by periodic normal subgroups, call G a generally finite group. Theorem 3 implies Corollary 1. If the set of finite order elements in a generally finite group G i...

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“…In this article we generalize some results of [1][2][3][4][5][6][7][8][9][10] and prove the theorems that are announced in [11,12].…”

mentioning

confidence: 83%

“…

…”

mentioning

confidence: 92%

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confidence: 99%

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On the existence of f-local subgroups in a group

Sozutov¹,

Yanchenko²

2006

Sib Math J

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“…Assertion 1) is established for the nontrivial full part of the group D similarly to Lemma 3.5 from [2]. We therefore assume that at least one of the subgroups A and B, for instance A, has a nontrivial full part, and D contains no nonidentity full subgroups.…”

Section: In This Case Either G Has General;~ed Chernlkov Periodic Pmentioning

confidence: 96%

“…Here we prove a result that enables us to generalize the theorem of [1] to the case of arbitrary periodic groups. In the proof we use the Shlmkov method from [2], and the main result of the present paper is the generalization of Theorem 3.1 in [2]. Condition 1) cannot be omitted in the theorem, because without this condition the theorem fails, in view of the ex~.mples of groups due to Novikov-Adyan [3] and 01'shanskii [4].…”

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Characterization of generalized Chernikov groups among groups with involutions

Senashov

1997

Math Notes

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ABSTRACT. The class of generalized Chernikov groups is characterized, i.e., the class of periodic locally solvable groups with the primary ascending chain condition. The name of the class is related to the fact that the structure of such groups is close to that of Chernikov groups. Namely, a Chernikov group is defined as a finite extension of a direct product of finitely many quasi-cyclic groups, and a generalized Chernikov group is a layer-finite extension of a direct product A of quasi-cyclic p-groups with finitely many factors for each prime p such that each of its elements does not commute elementwise with only finitely many Sylow subgroups of A. A theorem that characterizes the generalized Chernikov groups in the class of groups with involution is proved.KEY WORDS: Chernikov group, generalized Chernikov group, quasi-cyclic group, layer-finite group, Sylow subgroup. w IntroductionA promising direction in group theory, which has been developing since the sixties, is the characterization of certain known classes of groups in other classes of groups, i.e., the study of weakest possible sufficient conditions under which the groups of a broader class belong to narrower classes of groups. Earlier the author studied, from this point of view, the class of layer-6n~te groups, i.e., groups in which the set of elements of any given order is finite, and the class of finite extensions of layer-finite groups. The next natural step in this direction is the description of the generalized Chernikov groups, which are extensions of a layer-~n~te group by means of a layer-flnlte group. Just this class is treated in the present paper. In [1] generalized Chernikov groups were characterized in the class of periodic groups without second-order elements. Here we prove a result that enables us to generalize the theorem of [1] to the case of arbitrary periodic groups. In the proof we use the Shlmkov method from [2], and the main result of the present paper is the generalization of Theorem 3.1 in [2]. Condition 1) cannot be omitted in the theorem, because without this condition the theorem fails, in view of the ex~.mples of groups due to Novikov-Adyan [3] and 01'shanskii [4].For the main notions used in the paper, see the monograph by S. N. Chernikov [5]. w Known results, definitions, and auxiliary assertionsIn this section we present known results, prove auxiliary assertions, and also recall the necessary definitions. Definition.A periodic almost locally solvable group with the primary ascending chain condition is called a generalized Chernikov group.The term "generalized Chernlkov group" first appeared in [6]. Its use can be justified by the fact that, by the Polovitskii theorem (see Proposition 1), a generalized Chernikov group G is the extension of a direct product A of quasi-cyclic p-groups with finitely many factors for each prime p by means of a locally normal group B, and each element of G does not commute elementwise with only finitely many Sylow primary subgroups of A. By comparison, a Chernikov group is a t~nlte extension of ...

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Synthesis and characterization of nanophasic LaCoO₃ powders

Armelao

Bandoli

Barreca

et al. 2002

Surface & Interface Analysis

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Nanophasic LaCoO 3 powders were synthesized from inorganic compounds both in molten chloride fluxes and by using a solution combustion synthesis procedure. The samples were prepared in air and underwent subsequent heat treatments in order to yield the perovskite crystalline phase. The composition and microstructure of the samples were investigated by XPS, x-ray-induced AES, x-ray diffraction and electron paramagnetic resonance. Further information was gained by magnetic measurements. Nanophasic LaCoO 3 powders with an average crystallite size of 35 nm were obtained.

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On primary elements in groups

Cited by 7 publications

References 10 publications

On the existence of f-local subgroups in a group

On the existence of f-local subgroups in a group

Characterization of generalized Chernikov groups among groups with involutions

Synthesis and characterization of nanophasic LaCoO₃ powders

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On primary elements in groups

Cited by 7 publications

References 10 publications

On the existence of f-local subgroups in a group

On the existence of f-local subgroups in a group

Characterization of generalized Chernikov groups among groups with involutions

Synthesis and characterization of nanophasic LaCoO3 powders

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Product

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Synthesis and characterization of nanophasic LaCoO₃ powders