Existence of a normal complement of a group in the multiplicative group of its group ring

Suppose G is the adjoint group of a locally nilpotent algebra over a ring K of characteristic 0 without zero-divisors. If the orders of the elements of G are not invertible in K, then the normalized multiplicative group of the group ring KG is a semidirect product of the torsion-free congruence subgroup and G.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7–8, pp. 884–889, July–August, 1991.     

On an action of the group ring of a group of field automorphisms

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 6, pp. 162–168, November–December, 1992.     

Decomposition numbers of the group

Translated from Algebra i Logika, Vol. 27, No. 5, pp. 535–561, September–October, 1988.     

On the frobenius group

It is proved that forl > 1 the permutational rank of a Frobenius group with kernel of nilpotency class 1 is greater than 4^i–1+1. Let p and q be prime numbers, q¦p – 1, 1 < j < q. For all such p, q, j one constructs Frobenius groups of order p^{j + 1}q with kernel of nilpotency class j and order p^j+1. For p=7, q=3, j =2 we obtain O. Yu. Shmidt's known example.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1021–1030, July–August, 1991.     

A group in a group

1. In an Abelian group, a module, or more generally a one-based group H, the only definable groups are the obvious ones: if G is interpretable in H, then it has a definable subgroup of finite index which is definably isomorphic to a quotient A/B, where A and B are definable subgroups of a Cartesian power of H. 2. In such a group the introduction of those quotient groups weakly eliminates imaginary elements. More generally, for a stable theory the existence of canonical real bases for complete types implies the elimination of imaginary elements. 3. A group which is interpretable in a one-based structure is one-based. The property of being one-based is preserved by interpretation for theories of finite rank but not in general.Translated from French.Translated from Algebra i Logika, No. 3, pp. 368–378, May–June, 1990.     

Fundamental group of a 3-manifold

Translated from Algebra i Logika, Vol. 29, No. 3, pp. 345–349, May–June, 1990.     

Admissible sets in group theory

Translated fromAlgebra i Logika, Vol. 31, No. 4, pp. 413–433, July–August, 1992.     

Generalized nilpotency of the multiplicative group of a group ring

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 9, pp. 1179–1183, September, 1989.     

On weakly-central group extensions

Some properties of weakly-central extensions for which the quotient group by the kernel is hypercentral are investigated. The existence of a hypercentral coradical in any such periodic group is proved and the complementability of some of its subgroups in the group is established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1017–1021, July–August, 1991.     

Three problems on the group of symplectomorphisms

Moscow State Pedagogical Correspondence Institute. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 1, pp. 66–68, January–March, 1992.     

Minimal permutation representation of Thompson's simple group

Translated from Algebra i Logika, Vol. 27, No. 5, pp. 562–580, September–October, 1988.     

Arithmetic of a group of points of a formal group

For a formal group of finite height over a non-ramified extension of a ring of p-adic integers we construct a system of generators of a formal module that is the generalization of the canonic Shafarevich basis and the system of Henniart generators for Lubin — Tate groups.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 9–23, 1991.     

Simplicity condition for the fundamental ideal of the modular group algebra of a locally finite group

A criterion for the simplicity of the fundamental ideal of the group ring of a locally finite group is given. With its help the simplicity of this ideal for the group PSL(2, F), where F is an infinite field that is a union of its finite subfields, is proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1088–1091, July–August, 1991.     

On frattini extensions of the projective special linear group

Barnaul. Translated fromSibirskii Matematicheskii Zhurnal, Vol. 32, No. 2, pp. 88–93, March–April, 1991.     

A test of balance of a locally compact group

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 1, pp. 127–129, Janaury–February, 1988.     

Cohomologies of the group COS mod 2

Investigation of the cohomologies of the group COS in the sense of Artin was begun by V. I. Arnol'd [1]. In this article is given exhaustive information about the cohomologies of the group COS with coefficients in Z₂.Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 4, No. 2, pp. 62–73, April–June, 1970.     

A mechanical connection on the group of volume-preserving diffeomorphisms

Voronezh State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 2, pp. 61–62, April–June, 1988.     

Semigroups of the basic affine lie group and their automorphisms

Omsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 4, pp. 59–64, July–August, 1992.     

Subgroups of the orthogonal group that contain a group of quasidiagonal matrices

In the group of real orthogonal matrices we study the lattice of subgroups containing a group of quasidiagonal orthogonal matrices of fixed type with at most one first-order block.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 73–80, 1979.