# Chernikov N. S.

### 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 *N* ⊲ *G*, *G*/*N* is residually finite if *R* \(\subseteq\) *N* \(\subseteq\) *L*.

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

Chernikov N. S., Khmelnitskii N. A.

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.

### 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.

### 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*.

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

Chernikov N. S., Putilov S. V.

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*.

### 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*.

### Quotient Groups of Groups of Certain Classes

Chernikov N. S., Trebenko D. Ya.

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*).

### 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.

### 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 group*G*=*AB* with nilpotent subgroups*A* and*B*.

### 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.

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

Chernikov N. S., Dovzhenko S. A.

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.

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

Chernikov N. S., Trebenko D. Ya.

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 *N* ⊴ *G*, 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 \(\mathfrak{Y} \supseteq \mathfrak{A}\) .

### Groups with incidence condition for noncyclic subgroups

Chechulin V. L., Chernikov N. S., Polovitskii Ya. D.

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.

### 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 }) = ⊘,*i* ≠*j* is a periodic hyper-Abelian group if the subgroups*G* _{j} are almost locally normal.

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

Chernikov N. S., Chernikov S. N.

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 W_{0}-) 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.

### 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

### Socle and socle-finite groups

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

### 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

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

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

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

Chernikov N. S., Petravchuk A. P.

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

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

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

### 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

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

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

### Factorization theorems for locally graded groups

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

### Product of almost-Abelian groups

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

### Groups that are factorable by extremal subgroups

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

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

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

### Groups with complemented commutants of proper subgroups

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