2019
Том 71
№ 11

# Chernikov N. S.

Articles: 29
Brief Communications (English)

### A Note on FC-Groups

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 288-289

Let G be an arbitrary FC-group, let R be its locally soluble radical, and let L/R = L(G/R). We prove that, for NG, G/N is residually finite if R $\subseteq$ N $\subseteq$ L.

Article (Russian)

### Locally Nilpotent Groups with Weak Conditions of π-Layer Minimality and π-Layer Maximality

Ukr. Mat. Zh. - 2002. - 54, № 7. - pp. 991-997

We investigate locally nilpotent groups with weak conditions of π-layer minimality and π-layer maximality.

Article (Russian)

### On Socle and Semisimple Groups

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 866-880

We prove a theorem that gives a large array of new counterexamples to the known Baer (1949) and S. Chernikov (1959) problems related to socle groups. All these counterexamples are semisimple groups. We also establish many new properties of locally subinvariant semisimple subgroups. In particular, using these properties, we prove that all almost locally solvable M′-groups are Chernikov groups.

Article (Russian)

### Factorization of Periodic Locally Solvable Groups by Locally Nilpotent and Nilpotent Subgroups

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1697-1717

We establish a series of new results concerning periodic locally solvable and finite solvable groups G = AB with locally nilpotent or nilpotent subgroups A and B.

Article (Russian)

### On π-Solvable and Locally π-Solvable Groups with Factorization

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 840-846

We prove that, in a locally π-solvable group G = AB with locally normal subgroups A and B, there exist pairwise-permutable Sylow π′- and p-subgroups A π′, A p and B π′, B p , p ∈ π, of the subgroups A and B, respectively, such that A π′ B π′ is a Sylow π′-subgroup of the group G and, for an arbitrary nonempty set σ $\subseteq$ π, $$\left( {\prod\nolimits_{p \in {\sigma }} {A_p } } \right)\left( {\prod\nolimits_{p \in {\sigma }} {B_p } } \right)\quad {and}\quad \left( {A_{{\pi }\prime } \prod\nolimits_{p \in {\sigma }} {A_p } } \right)\left( {B_{{\pi }\prime } \prod\nolimits_{p \in {\sigma }} {B_p } } \right)$$ are Sylow σ- and π′ ∪ σ-subgroups, respectively, of the group G.

Article (Russian)

### Properties of a Finite Group Representable as the Product of Two Nilpotent Groups

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 531-541

We establish a series of new properties of a finite group G = AB with nilpotent subgroups A and B.

Brief Communications (Russian)

### Quotient Groups of Groups of Certain Classes

Ukr. Mat. Zh. - 2000. - 52, № 8. - pp. 1141-1143

For an arbitrary variety $\mathfrak{X}$ of groups and an arbitrary class $\mathfrak{Y}$ of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with $\mathfrak{X}$ - and $\mathfrak{Y}$ -factors (respectively, is a residually $\mathfrak{Y}$ -group) if G possesses an invariant system with $\mathfrak{X}$ - and $\mathfrak{Y}$ -factors (respectively, is a residually $\mathfrak{Y}$ -group) and N $\mathfrak{X}$ (respectively, N is a maximal invariant $\mathfrak{X}$ -subgroup of the group G).

Article (Russian)

### On Periodic Locally Solvable Groups Decomposable into the Product of Two Locally Nilpotent Subgroups

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 965-970

We establish new results concerning various properties of a periodic locally solvable group G = A B with locally nilpotent subgroups A and B one of which is hyper-Abelian.

Article (Russian)

### On finite solvable groups decomposable into the product of two nilpotent subgroups

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 809–819

We establish a series of results concerning various properties of a finite solvable groupG=AB with nilpotent subgroupsA andB.

Article (Russian)

### On groups factorized by two subgroups with Chernikov commutants

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 396-402

We establish results concerning the almost solvability and other properties of groups factorized by two subgroups with finite or Chernikov commutants.

Article (Russian)

### Primary graded groups with complementable non-Frattini subgroups

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1324–1333

We describe primary graded groups (in particular, locally graded, RN-groups) with complementable non-Frattini subgroups.

Article (Russian)

### Quotient groups of locally graded groups and groups of certain Kurosh-Chernikov classes

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1545–1553

We establish the validity of the inclusion G/N∈ X for groups G ∈ X under certain restrictions on NG, where X is one of the following classes, the class of locally graded groups, the class of RI-groups, or the class $\hat P\mathfrak{Y}$ for a fixed group variety $\mathfrak{Y} \supseteq \mathfrak{A}$ .

Article (Russian)

### One condition of complementability in groups

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1417-1425

We consider groups satisfying the following condition: Any subgroup of such a group that can be complemented in a larger subgroup can also be complemented in the entire group. A complete description of such groups is obtained under some weak conditions of finiteness.

Article (Russian)

### Groups with incidence condition for noncyclic subgroups

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 533-539

We give a description of groups with incidence condition for noncyclic subgroups to within minimal noncyclic subgroups. We present a complete constructive description of locally graded groups (in particular, arbitrary locally finite groups) satisfying this condition.

Brief Communications (Russian)

### On groups factorizable in commuting almost locally normal subgroups

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 429-431

We prove that an RN-group (in particular, locally solvable) G =G 1 G 2 ...G n with G i and π(G i ) ∩ π(G j ) = ⊘,ij is a periodic hyper-Abelian group if the subgroupsG j are almost locally normal.

Article (Ukrainian)

### Complementability conditions for a periodic almost solvable subgroup in the group containing it

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 822–826

It is proved that if every prime Sylow subgroup of a periodic almost solvable (more generally, periodic W0-) subgroup H of a group G has a complement in G and if, moreover, H is at most countable and the set ?(H) is finite, the subgroup H itself possesses a complement in G.

Article (Ukrainian)

### Factorization of groups by means of commuting periodic subgroups with no elements of identical prime orders

Ukr. Mat. Zh. - 1991. - 43, № 10. - pp. 1429–1436

Article (Ukrainian)

### Socle and socle-finite groups

Ukr. Mat. Zh. - 1991. - 43, № 7-8. - pp. 1066–1069

Article (Ukrainian)

### Infinite locally finite simple groups with a nontrivial kernel of the centralizer of an elementary Abelian 2-subgroup

Ukr. Mat. Zh. - 1988. - 40, № 5. - pp. 668–670

Article (Ukrainian)

### Factorization of linear groups and groups which have a normal system with linear factors

Ukr. Mat. Zh. - 1988. - 40, № 3. - pp. 362-369

Article (Ukrainian)

### A characterization of periodic locally solvable groups whose Sylow subgroups are solvable or have a finite exponent

Ukr. Mat. Zh. - 1987. - 39, № 6. - pp. 761–767

Article (Ukrainian)

### Factorizations of groups of automorphisms of a finitely generated module over a commutative ring

Ukr. Mat. Zh. - 1987. - 39, № 5. - pp. 670–671

Article (Ukrainian)

### Properties of the normal closure of the center of an FC-subgroup B of group G=AB with Abelian subgroup A

Ukr. Mat. Zh. - 1986. - 38, № 3. - pp. 364–368

Article (Ukrainian)

### Simple locally finite groups with the minimality condition for 2-subgroups

Ukr. Mat. Zh. - 1983. - 35, № 2. - pp. 265—266

Article (Ukrainian)

### Factorization theorems for locally graded groups

Ukr. Mat. Zh. - 1982. - 34, № 6. - pp. 732—738

Article (Ukrainian)

### Product of almost-Abelian groups

Ukr. Mat. Zh. - 1981. - 33, № 1. - pp. 136–138

Article (Ukrainian)

### Groups that are factorable by extremal subgroups

Ukr. Mat. Zh. - 1980. - 32, № 5. - pp. 707–711

Article (Ukrainian)

### Best approximation of a function defined on a finite set

Ukr. Mat. Zh. - 1980. - 32, № 1. - pp. 133 - 140

Article (Ukrainian)

### Groups with complemented commutants of proper subgroups

Ukr. Mat. Zh. - 1979. - 31, № 1. - pp. 102–106