Non-archimedean group algebras

A logarithmic approach to classgroups of integral group rings

Class group of the group ring of a cyclic group of order p2 over the ring of integers in a quadratic number field

Clean group rings over localizations of rings of integers

On the codimension of modules over skew power series rings with applications to Iwasawa algebras

It is proved that from the isomorphism of the integral group rings of two groups follows the isomorphism of the quotient-groups of these groups by the second commutator subgroup. Groups with certain conditions T1 and T2 are considered. From condition T1 it follows that the group algebra over an integral domain has no zero-divisors, and from T2 that the group of invertible elements of this algebra is trivial. Sufficient conditions are given, under which there are no zero-divisors in the group algebra of a pro-p-group over a field of p elements.

It is shown that if a solvable group is not locally finite, then the group algebra over a field of characterisitc 0 has no nonzero algebraic ideals.

Let G be a tensor category over a field k of characteristic 0. In his paper [-i], w 7, Delinge had established rather general conditions for G to be the category of representations of an algebraic group. He has shown that if any object neC satisfies the condition A " n = 0 for a sufficiently large n, then 0 is equivalent to the category Rep(G) of representations of a proalgebraic group scheme G over k. For tensor categories over fields of positive characteristic this result is no longer true. In the first part of this paper we present a construction of a class of tensor categories over fields of positive characteristic for which an analogue of Deligne's theorem does not hold. To describe these categories, denote by Gz an algebraic Chevalley group over Z and by G the corresponding group over Fp (for a sufficiently large p). Our category C will be a certain "subquotient" of the category R(G) of finite-dimensional representations of G. More precisely, let S be the full subcategory of R(G) whose objects (called small representations) are direct sums of the irreducible representations of G with highest weights inside the fundamental alcove. Let also SL 2 c G be the principal SLz-subgroup in G and (SLz)Im c SL2 be the infinitesimal neighbourhood of the Cartan subgroup H in SL2. Denote by O the full subcategory of R(G) consisting of representations M of G (called quasiprojective representations) such that the restriction of M to (SL2)m) is projective. The category 0 is a quotient of the subcategory S+PcR(G). In the last section of the paper we show that the same approach can be applied to representations of quantum groups. Let U be the quantum group at a root of prime order p of 1 (which is a Hopf algebra over the ring of integers in a cyclotomic field, see [2, 3]). Starting from the category of representations of U, we construct a semisimple braided category ~(G, p) with the finite number of simple objects. On the other hand, in Wess-Zumino-Witten conformal field theory one encounters the category O(G, m) of integrable representations of the correspond-

This thesis consists of two independent chapters. The first chapter concerns the Smith Normal Form (SNF) over the integers ℤ of integral matrices. We consider the SNF of a matrix A to be the ratio of two ℤ-modules -- a finitely generated abelian group; this is called the Smith group of A. The Smith group provides a unified setting to present both new and old results. The new results concern the relationship between the eigenvalues of an integral matrix and its SNF. In particular, the multiplicities of integer eigenvalues are shown to relate to the multiplicities in the type of the Smith group. Bounds are also given for the exponent of the Smith group. In some cases, these are best possible. The old results discussed are the interlacing of the SNF in the case of augmented matrices and the symmetries of the SNF for certain combinatorial matrices. The latter results are extended to rectangular matrices. Numerous examples are given throughout, along with many conjectures based on computation. The second chapter generalizes the work of Pless, et al. on duadic codes and Q-codes. We take abelian group codes to be ideals in the group ring F[G], where G is a finite abelian group of odd order n and F is a finite field with characteristic relatively prime to n. We define generalized Q-codes from a pair of idempotents of F[G] and an automorphism of G which together obey two simple equations. These codes are (n, (n+1)/2) and (n, (n-1)/2) linear codes. We show that all of the properties of duadic and Q-codes generalize. In particular, we extend the results on the relationship of these codes to projective planes with regular automorphism group G. When F has characteristic 2, we give simple numerical conditions on G and F which determine when generalized Q-codes exist. We also give some techniques for constructing these codes.

Describing the group of units of a group ring is a classical problem. Let [Formula: see text] be a rational prime number. We set [Formula: see text] a primitive root of unity of order [Formula: see text], [Formula: see text] the ring of [Formula: see text]-cyclotomic integers, [Formula: see text] a finite abelian [Formula: see text]-group and [Formula: see text] the group of the units [Formula: see text] of [Formula: see text] such that [Formula: see text], where [Formula: see text] is the augmentation map. We will prove that all the elements of the group [Formula: see text] arise from the units of the group ring [Formula: see text], where [Formula: see text] is the cyclic group of order [Formula: see text]. As an application, we describe explicitly the group of units of the group ring [Formula: see text] when [Formula: see text] is an elementary abelian [Formula: see text]-group and [Formula: see text] is a regular prime number.

The motivation behind the material presented in these notes is the following question:- if N/K is a finite Galois extension of number fields with Galois group Γ, and if \( mathfrak{D}{\text{ and }}\mathfrak{D}N \) are the rings of algebraic integers in K and N respectively, then what can be said about \( mathfrak{D}N \) as a Γ-module? A complete answer to this would be a description of \( mathfrak{D}N \) as a module over the group ring \( mathfrak{D}\Gamma \) , but since in general \( mathfrak{D}N \) need not be free over \( mathfrak{D} \) , it is more fruitful to restrict scalars and view \( mathfrak{D}N \) as a ℤΓ-module.

Nilpotent decomposition in integral group rings

Let Q be the rational number field, K/Q be a finite Galois extension with the Galois group G, and let CK be the ideal class group of K in the wider sense. We consider CK as a G-module. Denote by I the augmentation ideal of the group ring of G over the ring of rational integers. Then CK/I(CK) is called the central ideal class group of K, which is the maximal factor group of CK on which G acts trivially. A. Fröhlich [3, 41 rationally determined the central ideal class group of a complete Abelian field over Q whose degree is some power of a prime. The proof is based on Theorems 3 and 4 of Fröhlich [2]. D. Garbanati [6] recently gave an algorithm which will produce the l-invariants of the central ideal class group of an Abelian extension over Q for each prime l dividing its order.

The simplest example of two non-isomorphic groups with the same character tables is provided by the non-abelian groups of order p3, p ≠ 2. Let G1 be the one of exponent p and let G2 be the other. If Q denotes the field of rational numbers, then Berman (2) has shown that QG1 ≈ QG2, where QGi denotes the rational group algebra. In this note we shall show that the corresponding statement is false for ZGi where Z is the ring of rational integers. More explicitly we shall show that ZG1 does not contain a unit of order p2 so that it is impossible to embed ZG2 in ZG1.

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G T=<1>,where G T is the inertial group of R.

In this paper, we continue our study of the tensor product structure of category of weight modules over a Hopf–Ore extensions of a group algebra kG, where k is an algebraically closed field of characteristic zero. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of χ and are different. Then we describe the Green ring of the tensor category . It is shown that is isomorphic to the polynomial algebra over the group ring in one variable when , and that is isomorphic to the quotient ring of the polynomial algebra over the group ring in two variables modulo a principle ideal when . When , is isomorphic to the quotient ring of a skew group ring modulo some ideal, where is a polynomial algebra over in infinitely many variables.