Nilpotency of the multiplicative group of a group ring

It is proven that if K is a commutative ring of characteristic p^m while group G contains p-elements, then the multiplicative group UKG of group ring KG is nilpotent if and only if G is nilpotent and its commutant G is a finite p-group. Those group algebras KG are described for which the nilpotency classes of groups G and UKG coincide.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 191–200, February, 1972.In conclusion, the author wishes to express her gratitude to A. A. Bovdi for his scientific direction.     

On group rings of abelian p-groups of any cardinality

The problem is studied of the connection between an Abelian p-group G of arbitrary cardinality and its group ring LG, where L is a ring with unity nonzero characteristic n0 (mod p), with p being a prime. In particular, it is shown that group ring LG defines to within isomorphism the basis subgroup of group G. If reduced Abelian p-group G has finite type and if its Ulm factors decompose into direct products of cyclic groups, then group ring LG determines group G to within isomorphism.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 381–392, October, 1969.     

Generators of the group of principal units of a cyclic p-extension of a regular local field

Suppose k is a local field that is an extension of the field of p -adic numbers of degree n and does not contain a primitive p -th root of 1, and suppose K/k is a cyclic p-extension with Galois group G. The group E of principal units of K is a multiplicatively written module over the group ring =_p[G], where _p is the ring of p-adic integers. It was shown by Borevich (Ref. Zh. Mat., 1965, 3A256) that the -module E has a system of n+1 generators, of which n–1 are free and two are connected by certain relations. In the present paper these -generators are constructed explicitly and their arithmetical characteristics indicated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 16–23, 1977.     

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.     

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.     

Garlands containing the general linear group over an intermediate field

Let K/k be a finite extension of fields with an intermediate subfield L, and let H = GL_L(K) be the general linear group of all L-linear invertible mappings of the vector space of the field K over L. It is proved that the subgroups lying between GL_K(K)^H and the normalizer of H in G, where G = GL_k(K), form a garland. Bibliography: 4 titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 34–41.     

The index of a net subgroup of a symplectic group over a Dedekind ring

An explicit formula is derived for the index of a net congruence subgroup of a symplectic group over a Dedekind ring. A classification of symplectic D-nets over a field is obtained along the way.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 74–86, 1978.     

Finitely additive set functions into a topological group

The properties of finitely additive set functions defined on a ring S with values in an Abelian topological group are studied. In particular, a condition under which the property of uniform absence of the escaping load (UAEL) is preserved under the extension of a set function from a ring R to an extending ring S R is established.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 847–854, June, 1977.     

Homotopy equivalent group representations and Picard groups op the burnside ring and the character ring

We are concerned with the homotopy theory of group representations and its relation to character theory and the theory of the Burnside ring. We combine the methods of tom Dieck — Petrie [4] and torn Dieck [3] to show that the canonical map from the J-group jO(G), a subquotient of the representation ring RO(G), into the Picard group of the rational representation ring is injective for p-groups G. Moreover we compute the order of the cokernel of this map. We show that the Picard group of the rational representation ring is a direct summand in the Picard group of the Burnside ring. Finally we compute the Picard groups if G is abelian and indicate a computation for general G.     

Product of a biprimary and A 2-decomposable group

Suppose a finite group G is the product of a subgroups A and B of coprime orders, and suppose the order of A is p ^a q^b, where p and q are primes, and B is a 2-decomposable group of even order. Assume that a Sylow p-subgroup P is cyclic. If the order of P is not equal to 3 or 7, then G is solvable. If G is nonsolvable and G contains no nonidentity solvable invariant subgroups, then G is isomorphic to PSL(2, 7) or PGL(2, 7).Translated from Matematicheskie Zametki, Vol. 23, No. 5, pp. 641–649, May, 1978.     

Parabolic extensions of a Hecke ring of the general linear group

One describes a wide class of polynomials with coefficients from a Hecke ring of the general linear group, decomposing into factors over the Hecke ring of an appropriate parabolic subgroup. One gives examples of such decompositions, used in the theory of modular forms, and also the decomposition of the denominator of the Hecke series of the general linear group, yielding another proof of the rationality of the Hecke series of the group GL_n.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 154, pp. 36–45, 1986.     

On the projective dimension of torsion-free abelian groups over the ring of its endomorphisms

Let E be the ring of endomorphisms of an abelian group G, and let the group G be given the structure of a left module over the ring E in the natural way. We solve the problem of determining the projective dimension of the module.Translated from Matematicheskie Zametki, Vol. 7, No. 1, pp. 117–124, January, 1970.     

On the splitting of groups of central group extensions

It is proved that if G and V are abelian groups, then the group Ext(G, V) of all abelian extensions of the group V with the aid of the group G is extracted by the direct component in the group Opext(G, V) of all central extensions of V with the aid of G.Translated from Matematicheskie Zametki, Vol. 6, No. 2, pp. 197–200, August, 1969.     

Congruence subgroups of the symplectic group

Congruence subgroups closely related to parabolic subgroups are described in the symplectic group over a Dedekind ring.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 80–91, 1976.     

Moduli of continuity of functions,defined on a zero-dimensional group

It is shown that the condition is a criterion of modulus of continuity in the spaces C(G), L(G), and L₂(G) of functions defined on a zero-dimensional compact Abelian group G.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 379–388, March, 1978.     

The ring of integers of an Abelian extension of an algebraic number field as a Galois module

The ringO of integers of a finite Abelian extension K of an algebraic number field k is studied as a module over the group ring =[G], where is the ring of integers of k and G is the Galois group of K/k. It is proved that the ring is a decomposable -module if and only if there exists in K/k an intermediate extension K/F. FK, whose degree divides the different.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta. im. V. A. Steklova AN SSSR, Vol. 71, pp. 80–84, 1977.     

Elementary representations of the laguerre group

A class of representations of the Laguerre group in nuclear spaces is studied. The Laguerre group is the group of matrices of order two with determinant 1 over the ring of dual numbers. The question of irreducibility is considered, and a classification of bilinear invariant functionals, intertwining operators, and Hermitian invariant functionals is obtained.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 31–39, January, 1978.     

The automorphism group of a product of hypergraphs

A theorem of G. Sabidussi (1959, Duke Math. J. 26, 693–696) gives necessary and sufficient conditions for the automorphism group of the wreath product of two graphs to be the wreath product of their respective automorphism groups. In this paper we define a wreath product of hypergraphs and prove a theorem extending that of Sabidussi.