# Volume 54, № 7, 2002

### Groups with Maximality Condition for Nonhypercentral Subgroups

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

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

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

Ukr. Mat. Zh. - 2002. - 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.

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

Levchuk V. M., Voitenko T. Yu.

Ukr. Mat. Zh. - 2002. - 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*.

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

Dokuchaev M. A., Kirichenko V. V.

Ukr. Mat. Zh. - 2002. - 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.

### Modules over Group Rings with Certain Finiteness Conditions

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

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

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

Ukr. Mat. Zh. - 2002. - 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.

### On Solvable Normal Subgroups of Finite Groups

Gribovskaya E. E., Monakhov V. S., Sel’kin V. M.

Ukr. Mat. Zh. - 2002. - 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.

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

Ukr. Mat. Zh. - 2002. - 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.

### On Noetherian Modules over Minimax Abelian Groups

Ukr. Mat. Zh. - 2002. - 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.

### On a Hall Hypothesis

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

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

### 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.

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

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

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

Alekseeva O. A., Kondratiev A. S.

Ukr. Mat. Zh. - 2002. - 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*).

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

Ukr. Mat. Zh. - 2002. - 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.

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

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

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

### Note on Symmetric Words in Metabelian Groups

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

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

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

Gorbachuk E. L., Maturin Yu. P.

Ukr. Mat. Zh. - 2002. - 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.

### On 2-Symmetric Words in Nilpotent Groups

Ukr. Mat. Zh. - 2002. - 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.

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

Ukr. Mat. Zh. - 2002. - 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 dim*I*/*A* <. We show that the condition of local finiteness of *L* is essential in this statement.