2019
Том 71
№ 11

### All Issues

Article (Russian)

### Groups with Maximality Condition for Nonhypercentral Subgroups

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 887-896

We obtain a characterization of not finitely generated groups with maximality condition for nonhypercentral (respectively, nonnilpotent) subgroups.

Article (Russian)

### On New Types of ω-Fibered Fitting Classes of Finite Groups

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 897-906

We construct an infinite set of new types of ω-fibered Fitting classes of finite groups that differ from ω-local classes. We also describe the structure of maximal inner ω-satellites for the main types and establish a relationship between ω-fibered and Ω-foliated Fitting classes.

Article (Ukrainian)

### Zenkov Hypothesis on Pair $р$-Sylow Intersections of Chevalley Groups of Characteristic $р$

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 907-918

We present all Chevalley groups over a finite field of arbitrary characteristic p in which the index of the normalizer of any pair of Sylow p-subgroups is relatively prime with p.

Article (English)

### Quasi-Frobenius Rings and Nakayama Permutations of Semiperfect Rings

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 919-930

We say that ${\mathcal{A}}$ is a ring with duality for simple modules, or simply a DSM-ring, if, for every simple right (left) ${\mathcal{A}}$ -module U, the dual module U* is a simple left (right) ${\mathcal{A}}$ -module. We prove that a semiperfect ring is a DSM-ring if and only if it admits a Nakayama permutation. We introduce the notion of a monomial ideal of a semiperfect ring and study the structure of hereditary semiperfect rings with monomial ideals. We consider perfect rings with monomial socles.

Article (Russian)

### Modules over Group Rings with Certain Finiteness Conditions

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 931-940

We study modules over the group ring DG all proper submodules of which are finitely generated as D-modules.

Article (Russian)

### On the Nonexistence of Strongly Regular Graphs with Parameters (486, 165, 36, 66)

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 941-949

We prove that a strongly regular graph with parameters (486, 165, 36, 66) does not exist. Since the parameters indicated are parameters of a pseudogeometric graph for pG 2(5, 32), we conclude that the partial geometries pG 2(5, 32) and pG 2(32, 5) do not exist. Finally, a neighborhood of an arbitrary vertex of a pseudogeometric graph for pG 3(6, 80) is a pseudogeometric graph for pG 2(5, 32) and, therefore, a pseudogeometric graph for the partial geometry pG 3(6, 80) [i.e., a strongly regular graph with parameters (1127, 486, 165, 243)] does not exist.

Article (Russian)

### On Solvable Normal Subgroups of Finite Groups

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 950-960

We consider solvable invariant subgroups of a finite group with bounded primary indices of maximal subgroups. We establish that an invariant subgroup of this type belongs to the product of classical formations and investigate its dispersibility.

Article (Ukrainian)

### Abstract Characteristic of the Class of Unitary Positional Algebras of Operations

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 961-968

We find an abstract characteristic of the class of unitary positional algebras of operations, i.e., algebras that contain a complete collection of selectors.

Article (Russian)

### On Noetherian Modules over Minimax Abelian Groups

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 969-980

We consider modules over minimax Abelian groups. We prove that if A is an Abelian minimax subgroup of the multiplicative group of a field k and if the subring K of the field k generated by the subgroup A is Noetherian, then the subgroup A is the direct product of a periodic group and a finitely generated group.

Article (Russian)

### On a Hall Hypothesis

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 981-990

We obtain a new criterion for the solvability of a finite group with a given family of Hall subgroups.

Article (Russian)

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

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 991-997

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

Anniversaries (Ukrainian)

### Volodymyr Oleksandrovych Marchenko (On His 80th Birthday)

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 997

Brief Communications (Russian)

### On Recognizability of the Group $E_8(q)$ by the Set of Orders of Elements

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 998-1003

We prove that if a finite group G has the same set of orders of elements as the group E 8(q), then O 3(G/F(G)) is isomorphic to E 8(q).

Brief Communications (Russian)

### On Finite $A$-Groups with Complementable Nonmetacyclic Subgroups

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 1004-1007

We study groups G satisfying the following conditions:

(i) G is a finite solvable group with nonidentity metacyclic second derived subgroup;

(ii) all Sylow subgroups of G are Abelian, but not all of them are elementary Abelian.

We give a description of the structure of such groups with complementable nonmetacyclic subgroups.

Brief Communications (English)

### Sections of Angles and $n$th Roots of Numbers

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 1008-1012

It is known since Galois that an algebraic equation can be solved using suitable nth roots whenever the corresponding Galois group is soluble. The object of this note is to construct real numbers with the use of the nth parts of suitable angles and to state necessary and sufficient conditions for this to be possible.

Brief Communications (English)

### Note on Symmetric Words in Metabelian Groups

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 1013-1015

We completely describe n-symmetric words in a free metabelian group.

Brief Communications (English)

### $I$-Radicals, Their Lattices, and Some Classes of Rings

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 1016-1019

We describe some I-radicals in the categories of modules over semilocal rings. We give a characterization of rings over which the set of I-radicals coincides with the set of hereditary idempotent radicals. We prove that the lattices of I-radicals in the categories of modules over Morita-equivalent rings are isomorphic.

Brief Communications (English)

### On 2-Symmetric Words in Nilpotent Groups

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 1020-1024

We find the nilpotency class of a group of 2-symmetric words for free nilpotent groups, free nilpotent metabelian groups, and free (nilpotent of class c)-by-Abelian groups.

Brief Communications (Russian)

### On Strongly Inert Subalgebras of an Infinite-Dimensional Lie Algebra

Ukr. Mat. Zh. - 2002νmber=11. - 54, № 7. - pp. 1025-1028

We study infinite-dimensional Lie algebras L over an arbitrary field that contain a subalgebra A such that dim(A + [A, L])/A < ∞. We prove that if an algebra L is locally finite, then the subalgebra A is contained in a certain ideal I of the Lie algebra L such that dimI/A <. We show that the condition of local finiteness of L is essential in this statement.