# Volume 44, № 6, 1992

### A connection between quasi-finitariness and regularity in problems of semi-infinite linear programming

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 725–729

We generalize the approach, introduced by S. N. Chernikov, that reduces nonhomogeneous systems to homogeneous ones, and apply this approach, in particular, in a study of a class of finitary-definite problems. The generalization consists of such a modification as the approach for a more general class of problems, namely quasi-finitary problems, including the irregular case.

### Cohn's embedding of an enveloping algebra into a division ring

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 729–735

Let F be a field, L a Lie F-algebra and U=U(L) the universal enveloping algebra of L. In [1] Cohn constructs an embedding of U into a division ring. Recently there has been interest in this specific division ring in connection with matrix groups and matrix rings [2–4]. Cohn's construction is less than direct and it seemed useful to have a very explicit description of D, at least for the benefit of group theorists.

### The second brauer theorem

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 736–741

The inverse of the well-known Brauer formula is given, which is a consequence of his second main theorem. The formula can be applied to an evaluation of values of characters of local subgroups. Moreover, if b is a block of C_{G}(?), B is a block of a group G, then a method is found to check the equality b^{G}=B.

### Chernikov p-groups and integral p-adic representations of finite groups

Drobotenko V. S., Gudivok P. M., Vashchuk F. G.

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 742–753

The connection is studied between Chernikov p-groups and integral p-adic representations of finite p-groups. A description is presented with a precision up to isomorphism of certain classes of Chernikov p-groups.

### Representations of finite p-groups over the ring of formal power series with integral p-adic coefficients

Gudivok P. M., Oros V. M., Roiter A. V.

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 753–765

### Symmetric duality for lexicographic linear programming problems

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 766–773

A symmetric pair of lexicographic linear-programming (LP) problems, connected by regular relations, is formulated for problems of multicriteria linear optimization. A duality theorem for improper linear-programming problems (ILPP) is constructed in terms of lexicographic optimization.

### Products of groups with cyclic sylow p-subgroups and groups with nontrivial centers

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 773–779

This article considers finite groups that can be decomposed into the product of two proper subgroups.

### Finite p-groups with complemented maximal cyclic subgroups

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 780–785

### Commutator structure of some subgroups of chevalley groups

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 786–795

Question 6.34 of the Kourov Notebook is solved: carry the familiar theorem of Merzlyakov on the joint commutator subgroup of covering subgroups over to Chevalley groups and moreover to orthogonal and unitary groups.

### Groups all of whose infinite abelian pd-subgroups are normal

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 796–800

The author studies groups in which any infinite Abelian pd-subgroup (p is a prime) is normal, on the assumption that the group indeed contains such subgroups (IH_{p}-groups). Necessary and sufficient conditions are established for a group to be an IH_{p}-group. Relationships are established between the class of IH_{p}-groups and the class of groups in which all infinite Abelian subgroups are normal, and the class of groups in which all pd-subgroups are normal.

### Convolution method and informativeness of the recognition space

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 800–805

We give a short survey of application of linear inequalities, in particular of Chernikov's convolution method, to various problems of pattern recognition, estimation of the informativeness of spaces of the test description of the classification objects, etc.

### Sufficient conditions for extremum, penalty functions and regularity

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 805–812

General conditions for regularity of a point relative to a system of nonlinear equations are stated. It is shown that if the regularity conditions are fulfilled, the general problem of mathematical programming could be reduced to the minimization of a nonsmooth penalty function and the necessary conditions for the extremum could be stated in the most general form.

### ?-Free groups as groups with length function

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 813–822

It is shown that there exists a length function with values in a finitely generated group ? relative to which G is a ?-free group in any finitely generated group G.

### Constructive description of finite nondispersive groups in which all subgroups of composite index are abelian

Chernikov S. N., Levishchenko S. S.

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 818–822

A theorem which provides a constructive description of finite nondispersive groups in which all subgroups of composite index are Abelian is proved.

### 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.

### Sufficient criterion for the existence of a 2-complete part of a group

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 827–835

We obtain a sufficient condition for the existence of a 2-complete part of a group.

### Groups with finite rank elements

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 836–839

With the aid of the notion of the rank of an element in an arbitrary group, we prove a criterion for an infinite group to be nonsimple and find conditions under which a q-biprimitively finite group G with Chernikov Sylow q-subgroups has a Chernikov quotient group G/O_{p?}(G).

### Locally graduated groups with complemented infinite nonprimary subgroups

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 839–842

The investigations of infinite groups with certain systems of complemented infinite subgroups, suggested by Chernikov, are continued. It is proved that an infinic locally graduated, nonprimary group with complemented infinite nonprimary subgroups is locally finite and solvable, and all of its nonprimary subgroups have complements if and only if it is not Chernikov.

### Residual finiteness of descending HNN-extension of groups

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 842–845

The paper examines the special case of the general construction of HNN-extensions of groups in which at least one of the associated subgroups is the base group. A criterion is determined for a group obtained in this way to be residually finite. Any group obtained as such an extension from a free nilpotent group of finite rank is residually finite.

### Imbedding of periodic groups in simple periodic groups

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 845–847

It is proved that each periodic group is isomorphic to a subgroup of some simple periodic group.

### Groups with the layer-minimal condition

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 847–851

### Existence of a normal complement of a Hall subgroup

Romanovskii A. V., Sementovskii A. V.

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 851–853

### Major subgroups of nilpotent-by-finite groups

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 853–856