UQ eSpace

A theorem of K. W. Roggenkamp and L. L. Scott shows that for a finite group G with a normal p-subgroup containing its own centralizer, any two group bases of the integral group ring ZG are conjugate in the units of ZpG. Though the theorem presents itself in the work of others and appears to be needed, there is no published account. There seems to be a flaw in the proof, because a ‘theorem’ appearing in the survey [K. W. Roggenkamp, ‘The isomorphism problem for integral group rings of finite groups’, Progress in Mathematics 95 (Birkhäuser, Basel, 1991), pp. 193--220], where the main ingredients of a proof are given, is false. In this paper, it is shown how to close this gap, at least if one is only interested in the conclusion mentioned above. Therefore, some consequences of the results of A. Weiss on permutation modules are stated. The basic steps of which any proof should consist are discussed in some detail. In doing so, a complete, yet short, proof of the theorem is given in the case that G has a normal Sylow p-subgroup. 2000 Mathematical Subject Classification: primary 16U60; secondary 20C05.

A ring is right finite dimensional if it contains no infinite direct sum of right ideals.We prove that if a group G is finite, free abelian, or finitely generated abelian, then a ring R is right finite dimensional if and only if the group ring RG is right finite dimensional.A ring R is a self-injective cogenerator ring if Rn is injective and RR is a cogenerator in the category of unital right /{-modules; this means that each right unital -module can be embedded in a direct product of copies of R. Let G be a finite group where the order of G is a unit in R. Then the group ring RG is a selfinjective cogenerator ring if and only if R is a self-injective cogenerator ring.Additional applications are given.

In this paper, for any Milnor hypersurface, we find the largest dimension of effective algebraic torus actions on it. The proof of the corresponding theorem is based on the computation of the automorphism group for any Milnor hypersurface. We find all generalized Buchstaber–Ray and Ray hypersurfaces that are toric varieties. We compute the Betti numbers of these hypersurfaces and describe their integral singular cohomology rings in terms of the cohomology of the corresponding ambient varieties.

Let e: 1 -> N - G - K -> 1 be an extension of a finite cyclic group N by a finite cyclic group K .Using homological perturbation theory, we introduce the beginning of a free resolution of the integers Z over the group ring ZG of G in such a way that the resolution reflects the structure of G as an extension of N by K, and we use this resolution to compute the additive structure of the integral cohomology of G in many cases.We proceed by first establishing a number of special cases, thereafter constructing suitable cohomology classes thereby obtaining a lower bound, then computing characteristic classes introduced in an earlier paper, and, finally, exploiting these classes, obtaining upper bounds for the cohomology via the integral cohomology spectral sequence of the extension e.The calculation is then completed by comparing the two bounds.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Nilpotent decomposition in integral group rings

Variants of some of the Brauer-Fowler theorems

This paper gives the cohomology classification of finitistic spaces X equipped with free actions of the group G = S3 and the cohomology ring of the orbit space X/G is isomorphic to the integral cohomology quaternion projective space HPn. We have proved that the integral cohomology ring of X is isomorphic either to S4n+3 or S3 ? HPn. Similar results with other coefficient groups and for G = S1 actions are also discussed. As an application, we determine a bound of the index and co-index of cohomology sphere S2n+1 (resp. S4n+3) with respect to S1-actions (resp. S3-actions).

Given an affine surface X with rational singularities and minimal resolution X′, the covering of the Artin component of the deformation space of X where simultaneous resolutions are achieved is Galois and the Galois group is the Weyl group W associated with the configuration of (−2)-curves on X′. This gives the existence of actions of W on polynomial rings over Z where the ring of invariants is also polynomial. In turn, this leads to a description of the integral cohomology rings of flag varieties of type ADE that extends the known description of the rational cohomology rings as rings of coinvariants for actions of W.

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

The profile of a relational structure $R$ is the function $\phi_R$ which counts for every integer $n$ the number $\phi_R(n)$, possibly infinite, of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\phi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $KA(R)$, introduced by P. J. Cameron. In this paper we give a closer look at this association, particularly when the relational structure $R$ admits a finite monomorphic decomposition. This setting still encompass well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. We prove that $\phi_R$ is eventually a quasi-polynomial, this supporting the conjecture that, under mild assumptions on $R$, $\phi_R$ is eventually a quasi-polynomial when it is bounded by some polynomial.

Ever since Graham Higman's notable thesis [4] the “isomorphism problem for integral group rings” has withstood many attacks:

Every module over an Iwanaga–Gorenstein ring has a Gorenstein flat cover [13] (however, only a few nontrivial examples are known). Integral group rings over polycyclic-by-finite groups are Iwanaga–Gorenstein [10] and so their modules have such covers. In particular, modules over integral group rings of finite groups have these covers. In this article we initiate a study of these covers over these group rings. To do so we study the so-called Gorenstein cotorsion modules, i.e. the modules that split under Gorenstein flat modules. When the ring is ℤ, these are just the usual cotorsion modules. Harrison [16] gave a complete characterization of torsion free cotorsion ℤ-modules. We show that with appropriate modifications Harrison's results carry over to integral group rings ℤG when G is finite. So we classify the Gorenstein cotorsion modules which are also Gorenstein flat over these ℤG. Using these results we classify modules that can be the kernels of Gorenstein flat covers of integral group rings of finite groups. In so doing we necessarily give examples of such covers. We use the tools we develop to associate an integer invariant n with every finite group G and prime p. We show 1≤n≤|G : P| where P is a Sylow p-subgroup of G and gives some indication of the significance of this invariant. We also use the results of the paper to describe the co-Galois groups associated to the Gorenstein flat cover of a ℤG-module.

In this article, it is shown that the normalizer property holds for the following two kinds of finite nilpotent-by-nilpotent groups: (1) G = NwrH is the standard wreath product of N by H, where N is a finite nilpotent group and H is a finite abelian 2-group; (2) G is a finite group having a normal nilpotent subgroup N such that the integral group ring ℤ(G/N) has only trivial units. Our results generalize a result of Yuanlin Li and extend some ones obtained by Juriaans, Miranda, and Robério.

Let G be a finite group and X a space on which G acts. In this chapter we will describe a cohomological analysis of X which involves H*(G; \({\Bbb F}\) p) in a fundamental way. First developed by Borel and then by Quillen, this approach is the natural generalization of classical Smith Theory. After reviewing the basic constructions and a few examples, we will apply these techniques to certain complexes defined from subgroups of a group G, first introduced by K. Brown and D. Quillen. Using results due to Brown and P. Webb we will show how these complexes provide a systematic method for approaching the cohomology of simple groups which will be discussed later on. One of the aims of this chapter is to expose the reader to the important part played by group cohomology in the theory of finite transformation groups. By no means is this a complete account; in addition it requires a different background than the previous chapters have. We recommend the texts [AP], [Brel] as excellent sources for those wanting to learn more about group actions. For a more recent survey, we suggest [AD].KeywordsFinite GroupSpectral SequenceSimplicial ComplexEquivariant CohomologyHomotopy EquivalentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.