Zassenhaus conjecture for A 6

For the alternating group A ₆ of degree 6, Zassenhaus’ conjecture about rational conjugacy of torsion units in integral group rings is confirmed. Dedicated to the memory of I. S. Luthar Professor Luthar died at the age of 74 in December 2006.     

Vaught's Conjecture and Group Rings

We discuss Vaught's conjecture for complete theories of modules over some group rings over the integers and other Dedekind domains, and over pullback rings of Dedekind domains.     

Bifurcation analysis in a diffusive Segel–Jackson model

Bifurcations of spatially nonhomogeneous periodic orbits and steady state solutions are rigorously proved for a reaction–diffusion system modeling predator–prey interaction. The existence of these patterned solutions shows the richness of the spatiotemporal dynamics such as oscillatory behavior and spatial patterns.     

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.     

Central Units in Metacyclic Integral Group Rings

In this article, we give a method to compute the rank of the subgroup of central units of ? G, for a finite metacyclic group, G, by means of ?-classes and ?-classes. Then we construct a multiplicatively independent set  ? (U(? C _{p, q} )) and by applying our results, we prove that  generates a subgroup of finite index.     

Central Units in ℤCp,q

Let C_{p, q} be the semi-direct product of a cyclic group of order q by a cyclic group of order p, and ?C_{p, q} the integral group ring of C_{p, q}. In this article, firstly, we describe the group of normalized central units of ?C_{p, q} as a direct product of two subgroups that we call units of first kind and of second kind. For a class of prime numbers that we call good primes, we construct a multiplicatively independent set which generates the group of units of first kind. Finally, we construct a set of multiplicatively independent units which generates the units of second kind for a larger class of primes.     

K-theory of fine topological algebras, Chern character, and assembly

TheK-theory of the group algebra [] for a countable, discrete group is defined in terms of the simplicial ring of smooth simplices on [], where [] is given the fine topology with respect to its finite-dimensional, linear subspaces. The assembly map for this theory :K _* B K _* [] is studied and shown to be a rational injection. The proof uses the Connes-Karoubi Chern character fromK-theory of Banach algebras to cyclic homology, here generalized to any fine topological algebra, and proved to be multiplicative.     

FP-RINGS

A ring R is called right FP-injective if every R-homomor-phism from a finitely generated submodule of a free right R-module F into R extends to F. In this paper a ring R will be called a right FP-ring if R is semiperfect, right FP-injective and has an essential right socle. The class of FP-rings strictly contains the class of right (and left) pseudo-Frobenius rings, and we show that it is right-left symmetric and Morita-invariant. As an application we show that if R is a left perfect right FP-injective ring, then R is quasi-Frobenius if and only if the second right socle of R is finitely generated as a right ideal of R. This extends the known results in the right selfinjective case.     

On the Isomorphism Conjecture in algebraic K-theory

The Isomorphism Conjecture is a conceptional approach towards a calculation of the algebraic K-theory of a group ring RΓ, where Γ is an infinite group. In this paper we prove the conjecture in dimensions n<2 for fundamental groups of closed Riemannian manifolds with strictly negative sectional curvature and arbitrary coefficient rings R. If R is regular this leads to a concrete calculation of low dimensional K-theory groups of RΓ in terms of the K-theory of R and the homology of the group.     

Semiclean Rings

Abstract

The notion of semiclean elements in a ring is defined. Every clean element is semiclean. A ring R is said to be semiclean if every element in R is semiclean. The group ring Z _p G with G a cyclic group of order 3 is proved to be semiclean. The n × n matrix ring M _n (R) over a semiclean ring is semiclean. If R is a torsion free semiclean ring in which every element of R can be written as a sum of periodic and ±1, then R is clean. Every element in a semiclean ring R with 2 invertible is a sum of no more than 3 units.     

A character theoretic proof of the homological sylow theorem

ABSTRACT

Motivated by the works of Cuntz and Quillen on cyclic homology and algebra extensions, we study higher traces on group rings.     

Restricted Perfect Group Rings

A ring R is a restricted right perfect ring if every proper homomorphic image of R is right perfect. A complete characterization of restricted right perfect group rings RG has been obtained when the f.c. center of the group G is nontrivial. The f.c. center of a group G is the set of all elements of G that have finitely many conjugates in G.     

Uniform boundedness of torsion subgroups of linear groups

The torsion conjecture says： for any abelian variety A defined over a number field k, the order of the torsion subgroup of A（k） is bounded by a constant C（k,d） which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G（k）. Then the order of Г is bounded by a constant C＇（k, d） which depends only on k and the dimension d of G.