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.     

The equivariant Brauer group of a group

We consider the Brauer group of a group (finite or infinite) over a commutative ring with identity. A split exact sequence

is obtained. This generalizes the Fröhlich-Wall exact sequence from the case of a field to the case of a commutative ring, and generalizes the Picco-Platzeck exact sequence from the finite case of to the infinite case of . Here is the Brauer-Taylor group of Azumaya algebras (not necessarily with unit). The method developed in this paper might provide a key to computing the equivariant Brauer group of an infinite quantum group.

     

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.     

The permutation class group of a finite group

Analogously to the projective class group, the permutation class group of a finite group π can be defined as the group of equivalence classes of direct summands of integral permutation modules modulo permutation modules. It is shown that this group behaves nicely with respect to localization and completion, which then is used to prove that contrary to the projective class group - it is not always a torsion group. More precisely, the rank of the permutation class of group is computed.     

Abelian group pairs having a trivial coGalois group

Torsion-free covers are considered for objects in the category q ₂. Objects in the category q ₂ are just maps in R-Mod. For R = ℤ, we find necessary and sufficient conditions for the coGalois group G(A → B), associated to a torsion-free cover, to be trivial for an object A → B in q ₂. Our results generalize those of E. Enochs and J. Rado for abelian groups.     

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.     

Completion of the group topology of a countable group

Kishinev. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 1, pp. 3–13, January–February, 1990.     

The rational group algebra of a normally monomial group

In this paper, a complete irredundant set of a class of strong Shoda pairs of a finite group G is computed. The algebraic structure of the rational group algebra of a normally monomial group is thus obtained. A necessary and sufficient condition for G to be normally monomial is derived. The main result is also illustrated by computing a complete set of primitive central idempotents and the explicit Wedderburn decomposition of the rational group algebra of some normally monomial groups.     

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 the outer automorphism group of a hyperbolic group

LetG be a one-ended, word-hyperbolic group. Let Γ be an irreducible lattice in a connected semi-simple Lie group of rank at least 2. Ifh: Γ→Out(G) is a homomorphism, then Im(h) is finite. Dedicated to Professor Takushiro Ochiai for his sixtieth birthday     

Collineation group as a subgroup of the symmetric group

Let ψ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension ≥ 3 over a field. Let H be a closed (in the pointwise convergence topology) subgroup of the permutation group $\mathfrak{S}_\psi $ of the set ψ. Suppose that H contains the projective group and an arbitrary self-bijection of ψ transforming a triple of collinear points to a non-collinear triple. It is well known from [Kantor W.M., McDonough T.P., On the maximality of PSL(d+1,q), d ≥ 2, J. London Math. Soc., 1974, 8(3), 426] that if ψ is finite then H contains the alternating subgroup $\mathfrak{A}_\psi $ of $\mathfrak{S}_\psi $ . We show in Theorem 3.1 that H = $\mathfrak{S}_\psi $ , if ψ is infinite.     

On the group matrices for a generalized dihedral group

For the generalized dihedral group, an effective method is given to construct a matrix which transforms group matrices to a convenient block-diagonal form.     

On the group of polynomial functions in a group

Let G be a group and let n be a positive integer. A polynomial function in G is a function from G ⁿ to G of the form , where f(x ₁, . . . , x _n ) is an element of the free product of G and the free group of rank n freely generated by x ₁, . . . , x _n . There is a natural definition for the product of two polynomial functions; equipped with this operation, the set of polynomial functions is a group. We prove that this group is polycyclic if and only if G is finitely generated, soluble, and nilpotent-by-finite. In particular, if the group of polynomial functions is polycyclic, then necessarily it is nilpotent-by-finite. Furthermore, we prove that G itself is polycyclic if and only if the subgroup of polynomial functions which send (1, . . . , 1) to 1 is finitely generated and soluble.      

Every Cech-analytic Baire semitopological group is a topological group

Among other things, we prove the assertion given in the title. This solves a problem of Pfister.