Shunkov Groups Research Articles

Слойная конечность в некоторых группах

Infinite groups with finiteness conditions for an infinite system of subgroups are studied. Groups with a condition: the normalizer of any non-trivial finite subgroup is a layer-finite group or the normalizer of any non-trivial finite subgroup has a layer-finite periodic part are studied for beginning in the locally finite class of group, then in the class of periodic groups of Shunkov and finally in the class of Shunkov groups which are contain a strongly embedded subgroup with an almost layer-finite periodic part. The group $G$ is called the Shunkov group if for any prime $p$ and for every finite subgroups $H$ from $G$ any two conjugate elements of order $p$ from the factor-group $N_G(H)/H$ generate a finite subgroup. Results for almost layer-finite groups and groups with almost layer-finite periodic part are transferred to layer-finite groups and groups with layer-finite periodic part. New characterizations of layer-finite groups and groups with layer-finite periodic part are obtained.

О прямых произведениях групп диэдра в локально конечных группах

When studying infinite groups, as a rule, some finiteness conditions are imposed. For example, they require that the group be periodic, a Shunkov group, a Frobenius group, or a locally finite group. The concept of saturation allows us to effectively establish the internal structure of various classes of infinite groups. To date, a large array of results on groups saturated with various classes of finite groups has been obtained. Another important direction in the study of groups with saturation conditions is the study of groups saturated by direct products of various groups. Significant progress has been made in solving the problem of B. Amberg and L. S. Kazarin on periodic groups saturated with dihedral groups in the class of locally finite groups. It is proved that a locally finite group saturated by the direct product of a finite number of finite groups of a dihedron is isomorphic to the direct product of locally cyclic groups multiplied by an involution. It is also proved that a locally finite group saturated by the direct product of a finite number of finite dihedral groups is solvable.

Shunkov Groups Saturated with Almost Simple Groups

Shunkov Groups Saturated with Almost Simple Groups

Локально конечные группы, насыщенные прямым произведением двух конечных групп диэдра

In the study of infinite groups, as a rule, some finiteness conditions are imposed. For example, the group is required to be periodic, Shunkov group, Frobenius group, locally finite group. The concept of saturation makes it possible to effectively establish the internal structure of various classes of infinite groups. To date, a large array of results on groups saturated with various classes of finite groups has been obtained. Another important direction in the study of groups with saturation conditions is the study of groups saturated with direct products of various groups. In this paper, a partial solution to the problem of B. Amberg and L.S. Kazarin on periodic groups saturated with dihedral groups in the class of locally finite groups. The structure of a locally finite group saturated with a direct product of two finite dihedral groups is established and it is proved that in this case the group is solvable. The result obtained is an important step towards solving the problem of Amberg and Kazarin.

Shunkov Groups Saturated by General Linear Groups of Degree 3

Shunkov Groups Saturated by General Linear Groups of Degree 3

О двух свойствах группы Шункова

One of the interesting classes of mixed groups ( i.e. groups that can contain both elements of finite order and elements of infinite order) is the class of Shunkov groups. The group $G$ is called Shunkov group if for any finite subgroup $H$ of $G$ in the quotient group $N_G(H)/H$, any two conjugate elements of prime order generate a finite group. When studying the Shunkov group $G$, a situation often arises when it is necessary to move to the quotient group of the group $G$ by some of its normal subgroup $N$. In which cases is the resulting quotient group $G/N$ again a Shunkov group? The paper gives a positive answer to this question, provided that the normal subgroup $N$ is locally finite and the orders of elements of the subgroup $N$ are mutually simple with the orders of elements of the quotient group $G/N$. Let $ \mathfrak{X}$ be a set of groups. A group $G$ is saturated with groups from the set $ \mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $ G$ that is isomorphic to some group of $\mathfrak{X}$ . If all elements of finite orders from the group $G$ are contained in a periodic subgroup of the group $G$, then it is called the periodic part of the group $G$ and is denoted by $T(G)$. It is proved that the Shunkov group saturated with finite linear and unitary groups of degree 3 over finite fields has a periodic part that is isomorphic to either a linear or unitary group of degree 3 on a suitable locally finite field. \end{abstracte}

О периодических группах Шункова с почти слойно конечными нормализаторами конечных подгрупп

Layer-finite groups first appeared in the work by S. N. Chernikov (1945). Almost layer-finite groups are extensions of layer-finite groups by finite groups. The class of almost layer-finite groups is wider than the class of layer-finite groups; it includes all Chernikov groups, while it is easy to give examples of Chernikov groups that are not layer-finite. The author develops the direction of characterizing well-known and well-studied classes of groups in other classes of groups with some additional (rather weak) finiteness conditions. A Shunkov group is a group 𝐺 in which for any of its finite subgroups 𝐾 in the quotient group <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi>N</mi><mi>G</mi></msub><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mi>K</mi></mfrac></math> any two conjugate elements of prime order generate a finite subgroup. In this paper, we prove the properties of periodic not almost layer-finite Shunkov groups with condition: the normalizer of any finite nontrivial subgroup is almost layer-finite. Earlier, these properties were proved in various articles of the author, as necessary, sometimes under some conditions, then under others (the minimality conditions for not almost layer-finite subgroups, the absence of second-order elements in the group, the presence of subgroups with certain properties in the group). At the same time, it was necessary to make remarks that this property is proved in almost the same way as in the previous work, but under different conditions. This eliminates the shortcomings in the proofs of many articles by the author, in which these properties are used without proof.

On the periodic part of the Shunkov group saturated with linear groups of degree 2 over finite fields of even characteristic

On the periodic part of the Shunkov group saturated with linear groups of degree 2 over finite fields of even characteristic

О группах Шункова, насыщенных конечными простыми группами

О группах Шункова, насыщенных конечными простыми группами

Об одном достаточном условии существования периодической части в группе Шункова

Об одном достаточном условии существования периодической части в группе Шункова

О периодических группах и группах Шункова, насыщенных группами диэдра и А5

О периодических группах и группах Шункова, насыщенных группами диэдра и А5

Shunkov groups saturated with general linear groups

We prove that if a periodic Shunkov group is saturated with degree 2 general linear groups over finite fields then it is isomorphic to the degree 2 general linear group over a suitable locally finite field.

Shunkov Groups with the Minimal Condition for Noncomplemented Abelian Subgroups

Shunkov Groups with the Minimal Condition for Noncomplemented Abelian Subgroups

On Sylow Subgroups of Some Shunkov Groups

We study Shunkov groups with the following condition: the normalizer of any finite nontrivial subgroup has an almost layer-finite periodic part. Under this condition, we establish the structure of Sylow 2-subgroups in this group. Shunkov introduced a class of conjugate biprimitively finite groups. Since 2000, these groups are usually called Shunkov groups. In the present paper, we study Shunkov groups. A Shunkov group is defined as a group G such that, for any its finite subgroup H, any pair of conjugate elements of prime order generates a finite subgroup in the quotient group NG(H)/H. The following condition is imposed on the group: the normalizer of any nontrivial finite subgroup has an almost layer-finite periodic part. The class of groups satisfying this condition is rather wide. Thus, it contains Burnside groups of odd periods ≥ 665 [1] and Ol’shanskii groups [2]. A group is called layer-finite if the set of its elements of any given order is finite. This class was introduced by Chernikov in [3]. Almost layer-finite groups are defined as extensions of layer-finite groups with the help of finite groups. The class of almost layer-finite groups is much wider than the class of layer-finite groups. It contains all Chernikov groups. At the same time, there are Chernikov groups that are not layer-finite. Thus, in particular, this is true for the extension of a quasicyclic group with the help of reversing automorphism. If the product of all normal layer-finite subgroups of a group is layer-finite, then it is called a layer-finite radical of the group. The set of elements of finite order of a group is called a periodic part of the group if this set is a group. In Shunkov groups without almost layer-finite periodic parts such that the normalizer of any their nontrivial finite subgroup has an almost layer-finite periodic part, we study the structure of Sylow 2-subgroups. Earlier, we proved that if a Sylow 2-subgroup of the analyzed group is infinite, then it is the extension of a quasicyclic 2-group with the help of a reversing automorphism [4].

On the Shunkov groups acting freely on Abelian groups

We prove the existence of a locally finite periodic part in every rank 1 Shunkov group with solvable finite subgroups, in every Shunkov group acting freely on an abelian group, and in the groups of affine transformations of neardomains with finite elements.

On groups with a strongly imbedded subgroup having an almost layer-finite periodic part

We study Shunkov groups with the following condition: the normalizer of any finite nontrivial subgroup has an almost layer-finite periodic part. It is proved that such a group has an almost layer-finite periodic part if it has a strongly imbedded subgroup with almost layer-finite periodic part.

On Shunkov groups with a strongly embedded subgroup

Infinite Shunkov groups with the following condition are studied: the normalizer of any finite nontrivial subgroup has an almost layer-finite periodic part. Under this condition, the almost layer-finiteness is established of the periodic part of a Shunkov group with a strongly embedded subgroup having a Chernikov almost layer-finite periodic part. Earlier, the author proved the almost layer-finiteness of a Shunkov group with a strongly embedded subgroup under the conditions that all proper subgroups are almost layer-finite and that the group is periodic.

Characterizations of the Shunkov groups

We study the structure of a family of finite groups of the form L g = 〈a, a g 〉 in a periodic Shunkov group. As a consequence of the obtained result, we get two characterizations of periodic Shunkov groups.

A class of periodic groups

We deal with periodic groups saturated with dihedral groups. In particular, it is proved that periodic groups of bounded period, and also periodic Shunkov groups, saturated with dihedral groups, are locally finite.

Almost layer-finiteness of the periodic part of groups without involutions

We prove the following theorem which characterises the class of groups without involutions which have an almost layer-finite periodic part: if the normaliser of any non-trivial finite subgroup of a Shunkov group without involutions has an almost layer-finite periodic part, then the group also has an almost layer-finite periodic part.