Pregroups and the big powers condition

We study groups having the big powers property BP. It is proved that if a pregroup satisfies some natural axioms, then its universal group has this property. In particular, fundamental groups of some graphs of groups have the big powers property if BP holds for edge and vertex subgroups and a number of natural conditions are satisfied. The results obtained are applied to Lyndon’s completions U(P)^ℤ[t] of the universal group U(P) with P satisfying the conditions mentioned. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Translated from Algebra i Logika, Vol. 48, No. 3, pp. 342–377, May–June, 2009.     

Universal Central Extensions of Lie Groups

We call a central Z-extension of a group G weakly universal for an Abelian group A if the correspondence assigning to a homomorphism ZA the corresponding A-extension yields a bijection of extension classes. The main problem discussed in this paper is the existence of central Lie group extensions of a connected Lie group G which is weakly universal for all Abelian Lie groups whose identity components are quotients of vector spaces by discrete subgroups. We call these Abelian groups regular. In the first part of the paper we deal with the corresponding question in the context of topological, Fréchet, and Banach–Lie algebras, and in the second part we turn to the groups. Here we start with a discussion of the weak universality for discrete Abelian groups and then turn to regular Lie groups A. The main results are a Recognition and a Characterization Theorem for weakly universal central extensions.     

Equivalence and the Brauer–Clifford Group for G-Algebras over Commutative Rings

The notion of G-algebra equivalence for a group G is generalized from the setting of G-algebras over fields to G-algebras over commutative rings. This leads to a formulation of Turull's Brauer–Clifford group for separable G-algebras over commutative rings, and to connections with Fröhlich and Wall's equivariant Brauer group.     

Canonical and existential groups in universal classes of Abelian groups

Universal classes of Abelian groups are classified in terms of sets of finitely generated groups closed with respect to the discrimination operator. The notions of a principal universal class and a canonical group for such a class are introduced. For any universal class K, the class K^ec of existentially closed groups generated by the universal theory of K is described. It is proved that K^ec is axiomatizable and, therefore, the universal theory of K has a model companion.     

A New Characterization of PSL(p,q)

Abstract

Order components of a finite group are introduced in Chen [Chen, G. Y. (1996c) On Thompson's conjecture. J. Algebra 185:184–193]. It was proved that PSL(3, q), where q is an odd prime power, is uniquely determined by its order components [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002a). A characterization of PSL(3, q) where q is an odd prime power. J. Pure Appl. Algebra 170(2–3): 243–254]. Also in Iranmanesh et al. [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002b). A characterization of PSL(3, q) where q = 2 ⁿ . Acta Math. Sinica, English Ser. 18(3):463–472] and [Iranmanesh, A., Alavi, S. H. (2002). A characterization of simple groups PSL(5, q). Bull. Austral. Math. Soc. 65:211–222] it was proved that PSL(3, q) for q = 2 ⁿ and PSL(5, q) are uniquely determined by their order components. In this paper we prove that PSL(p, q) can be uniquely determined by its order components, where p is an odd prime number. A main consequence of our results is the validity of Thompson's conjecture for the groups under consideration.     

Equations and algebraic geometry over profinite groups

The notion of an equation over a profinite group is defined, as well as the concepts of an algebraic set and of a coordinate group. We show how to represent the coordinate group as a projective limit of coordinate groups of finite groups. It is proved that if the set π(G) of prime divisors of the profinite period of a group G is infinite, then such a group is not Noetherian, even with respect to one-variable equations. For the case of Abelian groups, the finiteness of a set π(G) gives rise to equational Noetherianness. The concept of a standard linear pro-p-group is introduced, and we prove that such is always equationally Noetherian. As a consequence, it is stated that free nilpotent pro-p-groups and free metabelian pro-p-groups are equationally Noetherian. In addition, two examples of equationally non-Noetherian pro-p-groups are constructed. The concepts of a universal formula and of a universal theory over a profinite group are defined. For equationally Noetherian profinite groups, coordinate groups of irreducible algebraic sets are described using the language of universal theories and the notion of discriminability.     

Diff(Rn) as a Milnor–Lie group

We describe a construction of the Lie group structure on the diffeomorphism group Diff( R ⁿ), modelled on the space D( R ⁿ, R ⁿ) of R ⁿ‐valued test functions on R ⁿ, in John Milnor's setting of infinite‐dimensional Lie groups. New tools are introduced to simplify this task. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Maximal Holonomy of Almost Bieberbach Groups for Heis5

Let Heis _2n+1 be the Heisenberg group of dimension 2n + 1 and M an infra-nilmanifold with Heis _2n+1-geometry. The fundamental group of M contains a cocompact lattice of Heis _2n+1 with index bounded above by a universal constant I _n+1, i.e., I _n+1 is the maximal order of the holonomy groups. We prove that I ₃ = 24. As an application we give an estimate for the volumes of finite volume non-compact complex hyperbolic 3-manifolds.     

Groups with Polycyclic-by-Finite Conjugate Classes of Subgroups

Abstract

Neumann characterized the groups in which every subgroup has finitely many conjugates only as central-by-finite groups. If 𝔛 is a class of groups, a group G is said to have 𝔛-conjugate classes of subgroups if G/Core _G (N _G (H)) ∈ 𝔛 for every subgroup H of G. In this paper, we generalize Neumann's result by showing that a group has polycyclic-by-finite classes of conjugate subgroup if and only if it is central-by-(polycyclic-by-finite).     

Transversals in non-discrete groups

The concept of ‘topological right transversal’ is introduced to study right transversals in topological groups. Given any right quasigroupS with a Tychonoff topologyT, it is proved that there exists a Hausdorff topological group in whichS can be embedded algebraically and topologically as a right transversal of a subgroup (not necessarily closed). It is also proved that if a topological right transversal(S, T _S ,T ^S , o) is such thatT _S =T ^S is a locally compact Hausdorff topology onS, thenS can be embedded as a right transversal of a closed subgroup in a Hausdorff topological group which is universal in some sense.     

An extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system

This paper proposes an extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitin's Ω is defined as the probability that the universal self‐delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program‐size complexity H (s) of a given finite binary string s. In the standard way, H (s) is defined as the length of the shortest input string for U to output s. In the other way, the so‐called universal probability m is introduced first, and then H (s) is defined as –log₂ m (s) without reference to the concept of program‐size. Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator‐valued measure (POVM) in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour‐El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi‐POVM. We also give another characterization of Chaitin's Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as a sum of the POVM elements of a universal semi‐POVM. The validity of this definition is discussed. In what follows, we introduce an operator version (s) of H (s) in a Hilbert space of infinite dimension using a universal semi‐POVM, and study its properties. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The axiomatic closure of the class of discriminating groups

In [6] squarelike groups were defined to be those groups G universally equivalent to their direct squares G × G. In that paper it was shown that G is squarelike if and only if G is universally equivalent to a discriminating group in the sense of [3]. Further it was shown that the class of squarelike groups is first-order axiomatizable while the class of discriminating groups is not. In this paper, we prove that the class of squarelike groups is the least axiomatic class containing the discriminating groups.Received: 18 August 2003     

Using Fox Derivatives in Treating Groups of the Form F/[R′, F]

For a factor group with respect to periodic part of a group of the form F/[R′, F], an embedding in the matrix group is defined. The criteria for a matrix to belong to an image of this group and for elements to be conjugate are specified. Some statements having a direct bearing on groups of the form in question are proved. Application of the results obtained allows us to refine the answer in [7] to a question by O. Chapuis concerning the universal classification of ∀-free soluble groups with two generators. Supported by RFBR grant No. 02-01-00293 and by FP “Universities of Russia” grant No. UR.04.01.227. __________ Translated from Algebra i Logika, Vol. 45, No. 1, pp. 114–125, January–February, 2006.     

Kirillov's Orbit Method for p-Groups and Pro-p Groups

In this text, we study Kirillov's orbit method in the context of Lazard's p-saturable groups when p is an odd prime. Using this approach we prove that the orbit method works in the following cases: torsion free p-adic analytic pro-p groups of dimension smaller than p, pro-p Sylow subgroups of classical groups over ? _p of small dimension and for certain families of finite p-groups.     

On the Prime Radical of PI-Representable Groups

The notion of PI-representable groups is introduced; these are subgroups of invertible elements of a PI-algebra over a field. It is shown that a PI-representable group has a largest locally solvable normal subgroup, and this subgroup coincides with the prime radical of the group. The prime radical of a finitely generated PI-representable group is solvable. The class of PI-representable groups is a generalization of the class of linear groups because in the groups of the former class the largest locally solvable normal subgroup can be not solvable.     

Integral Group Ring of the Mathieu Simple Group M 23

We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M ₂₃ using the Luthar–Passi method. This work is a continuation of the research that we carried out for Mathieu groups M ₁₁ and M ₁₂. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.     

Partially commutative metabelian groups: centralizers and elementary equivalence

For partially commutative metabelian groups, annihilators of elements of commutator subgroups are described; canonical representations of elements are defined; approximability by torsion-free nilpotent groups is proved; centralizers of elements are described. Also, it is proved that two partially commutative metabelian groups have equal elementary theories iff their defining graphs are isomorphic, and that every partially commutative metabelian group is embeddable in a metabelian group with decidable universal theory. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Supported by RFBR (project No. 09-01-00099). Translated from Algebra i Logika, Vol. 48, No. 3, pp. 309–341, May–June, 2009.     

An extension of Nori fundamental group

In this paper, we study a certain extension of Nori’s fundamental group in the case where a base field is of characteristic 0 and give structure theorems about it. As a result for a smooth projective curve with genus g>1, we prove that Nori’s fundamental group acts faithfully on the category of unipotent bundles on the universal covering. In the case when g = 1, we give a more finer result.     

Investigation of solvable (120, 35, 10) difference sets

Abelian difference sets with parameters (120, 35, 10) were ruled out by Turyn in 1965. Turyn's techniques do not apply to nonabelian groups. We attempt to determine the existence of (120, 35, 10) difference sets in the 44 nonabelian groups of order 120. We prove that if a solvable group admits a (120, 35, 10) difference set, then it admits a quotient group isomorphic to the cyclic group of order 24 or to U₂₄ ? 〈x,y : x⁸ = y³ = 1, xyx^?1 = y^?1〉. We describe a computer search, which rules out solutions with a ?₂₄ quotient. The existence question remains undecided in the three solvable groups admitting a U₂₄ quotient. The question also remains undecided for the three nonsolvable groups of order 120. © 2004 Wiley Periodicals, Inc.     

Multipliers for the symmetry groups of p-adic spacetime

The consequences for particle classification of the Volovich hypothesis that spacetime geometry is non-archimedean at the Planck scale are explored. The multiplier groups and universal topological central extensions of the p-adic Poincaré and Galilean groups are determined. The text was submitted by the author in English.