# Volume 54, № 6, 2002

### Sergei Nikolaevich Chernikov (on the 90th Anniversary of His Birth)

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 743-744

### On Subgroups of Prime Order in a Finite Group

Al-Sharo K. A., Shemetkov L. A.

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 745-752

We study finite groups with generalized normality condition for subgroups of prime order.

### Finitary and Artinian-Finitary Groups over the Integers $ℤ$

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 753-763

In a series of papers, we have considered finitary (that is, Noetherian-finitary) and Artinian-finitary groups of automorphisms of arbitrary modules over arbitrary rings. The structural conclusions for these two classes of groups are really very similar, especially over commutative rings. The question arises of the extent to which each class is a subclass of the other.

Here we resolve this question by concentrating just on the ground ring of the integers ℤ. We show that even over ℤ neither of these two classes of groups is contained in the other. On the other hand, we show how each group in either class can be built out of groups in the other class. This latter fact helps to explain the structural similarity of the groups in the two classes.

### Matrix Representations of Finite $p$-Groups over Commutative Local Rings of Characteristic $p^s$

Gudivok P. M., Pogorilyak E. Y.

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 764-770

We determine in what cases the problem of description of nonequivalent matrix representations of a finite $p$-group over a commutative local ring of characteristic $p^s$ is wild.

### A Presentation of the Automorphism Group of the Two-Generator Free Metabelian and Nilpotent Group of Class $c$

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 771-779

We determine the structure of IA(*G*)/Inn(*G*) by giving a set of generators, and showing that IA(*G*)/Inn(*G*) is a free abelian group of rank (*c* − 2)(*c* + 3)/2. Here *G* = *M* _{2}, *c* = 〈 *x*, *y*〉, *c* ≥ 2, is the free metabelian nilpotent group of class *c*.

### Groups with Various Minimal Conditions on Subgroups

Dixon M. R., Evans M. J., Smith H.

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 780-788

We briefly review some recent works on groups with the weak minimal condition on nonnilpotent subgroups. We also study the class of groups with the weak minimal condition on non-(soluble of derived length *d*) subgroups.

### Coxeter Functors for One Class of *-Quivers

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 789-797

For one certain class of *-quivers, we construct Coxeter functors and describe their application to the description of families of orthoprojectors whose sum is a multiple of the identity operator.

### Groups with Bounded Chernikov Conjugate Classes of Elements

Kurdachenko L. A., Otal J., Subbotin I. Ya.

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 798-807

We consider *BCC*-groups, that is groups *G* with Chernikov conjugacy classes in which for every element *x* ∈ *G* the minimax rank of the divisible part of the Chernikov group *G*/*C* _{G}(*x* ^{G}) and the order of the corresponding factor-group are bounded in terms of *G* only. We prove that a *BCC*-group has a Chernikov derived subgroup. This fact extends the well-known result due to B. H. Neumann characterizing groups with bounded finite conjugacy classes (*BFC*-groups).

### The Norm of a Relation, Separating Functions, and Representations of Marked Quivers

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 808-840

We consider numerical functions that characterize Dynkin schemes, Coxeter graphs, and tame marked quivers.

### Some Problems in Nonclassical Algebraic Geometry

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 841-846

We describe the general approach to a nonstandard geometry with the emphasis on associative algebras.

### Morphisms of Ball's Structures of Groups and Graphs

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 847-855

We introduce and study two kinds of morphisms between ball's structures related to groups and graphs.

### Minimality and Sylow-Permutability in Locally Finite Groups

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 856-865

We give a complete classification of the locally finite groups that are minimal with respect to Sylow-permutability being intransitive.

### 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.

### On Placement of Prime Order Elements in a Group

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 881-884

We characterize a class of *T* _{0}-groups related to the infinite Burnside groups of odd period.