Finite Group Rings Research Articles

The cohomology rings of finite groups with semi-dihedral Sylow 2-subgroups

In this paper we determine the mod-2 cohomology rings and the 2-primary part of the integral cohomology rings of finite groups with semi-dihedral Sylow 2-subgroups. The method used here is algebraic and can be considered as elementary.

A Sylowlike theorem for integral group rings of finite solvable groups

A Sylowlike theorem for integral group rings of finite solvable groups

On the isomorphism problem for integral group rings of finite groups

On the isomorphism problem for integral group rings of finite groups

On schur rings of group rings of finite groups

Schur rings are rings associated to certain partitions of finite groups. They were introduced for applications in representation theory, cfr. [3][4]. The algebric structure of these rings has not been studied in depth. In this paper we determine explicit structure constants for Schur rings, we derive conditions for separability and we compute the centre. These results seem to be new even over fields.

Construction of units in integral group rings of finite nilpotent groups

Let U be the group of units of the integral group ring of a finite group G. We give a set of generators of a subgroup B of U. This subgroup is of finite index in U if G is an odd nilpotent group. We also give an example of a 2-group such that B is of infinite index in U.

A dimension formula for the cohomology rings of finite groups with dihedral sylow 2-subgroups

A dimension formula for the cohomology rings of finite groups with dihedral sylow 2-subgroups

A categorical approach to the theory of equations

We construct a small category whose objects are monic square-free polynomials with coefficients in a field F. For a monic, irreducible, and normal polynomial, Aut ( f) is the usual Galois group of f. We prove that there exists a unique topological group G such that the category of finite discrete G-sets is equivalent to the opposite of our category. We then replace categories by commutative rings and define the Burnside ring of a field, which has Burnside rings of finite groups as its building blocks. We next extend scalars to the rationals and explicity determine the algebra that results. We find all valuations of this algebra and prove that an irreducible polynomial is completely determined by its values under these valuations.

Construction of units in integral group rings of finite nilpotent groups

Let U be the group of units of the integral group ring of a finite group G. We give a set of generators of a subgroup B of U. This subgroup is of finite index in U if G is an odd nilpotent group. We also give an example of a 2-group such that B is of infinite index in U.

The cohomology of the groups of order 32

We have calculated the mod - 2 \bmod \text {-}2 cohomology rings of all the groups of 32 elements. This paper describes the methods of calculation; the computer routines used can be adapted to assist in the calculation of the modular cohomology rings of other finite groups. We also describe the results of the calculations; the data we have collected provide a substantial increase in the supply of completed calculations in group cohomology, and so we take this opportunity to compare known results and open conjectures.

Invariants of group rings

The Schur group, uniform (distribution) group, Schur index, and Schur exponent are examined in the context of separable group rings of finite groups over commutative rings.

Addendum to “prine kernel functors in the skew group rings of finite groups”

Addendum to “prine kernel functors in the skew group rings of finite groups”

SK 1 of finite group rings: V

SK 1 of finite group rings: V

Prime kernel functors in the skew group rings of finite groups

Prime kernel functors in the skew group rings of finite groups

Some remarks on affine rings

Various topics on affine rings are considered, such as the relationship between Gelfand-Kirillov dimension and Krull dimension, and when a "locally affine" algebra is affine. The dimension result is applied to study prime ideals in fixed rings of finite groups, and in identity components of group-graded rings.

Cancellation over commutative rings of dimension one and two

All rings in this paper are commutative and Noetherian, and all modules are finitely generated. We will study the question: If A @ C z B @ C, is A z B? (Here A, B and C are modules over a ring R.) In 1962 Chase [4] gave an affirmative answer for R = k[X, Y], the polynomial ring in two variables over an algebraically closed field k, provided A and B are torsionfree and char(k)trank(A). In the first section of this paper we extend Chase’s result to include all two-dimensional regular affine k-domains. Most of the paper deals with one-dimensional rings. In Section 2 we prove some general results, which we hope will eventally lead to some sort of structure theory for torsionfree modules over one-dimensional reduced rings with finite normalization. In this context we show that one can cancel from projectives (that is, the answer is “yes” when A and B are projective) if and only if Pit R = Pit z, where g is the normalization. When A and B are assumed to be merely torsionfree, the problem is more subtle, but we have some partial results. The last three sections deal with three naturally occurring examples of one-dimensional rings: coordinate rings of curves, quadratic orders and integral group rings of finite abelian groups. In each of these cases we are able to give a fairly complete answer to the cancellation problem for torsionfree modules. Many of the ideas in this paper are the direct or indirect results of innumerable conversations with Raymond Heitmann and Lawrence Levy. In particular, Levy discovered a serious error in the proof of the main theorem of an earlier version, and Heitmann showed by example that the statement itself was incorrect. I am extremely grateful to both of them for their interest and insight.

On Fixed Rings of Automorphisms

In this note we present simple proofs of some well-known results on fixed rings of finite automorphism groups of associative rings.

Fixed point rings of finite automorphism groups of hereditary finitely generated P.I. algebras

Fixed point rings of finite automorphism groups of hereditary finitely generated P.I. algebras

On fixed rings of automorphisms

In this note we present simple proofs of some well-known results on fixed rings of finite automorphism groups of associative rings.

On the unit groups of Burnside rings of finite groups

On the unit groups of Burnside rings of finite groups

Strongly Graded Rings of Finite Groups

(1983). Strongly Graded Rings of Finite Groups. Communications in Algebra: Vol. 11, No. 10, pp. 1033-1071.