Klaus W. Roggenkamp

Isomorphisms of p-adic group rings

A long-standing problem, first posed by Graham Higman [15] and later by Brauer [4] is the “isomorphism problem for integral group rings.” Given finite groups G and H, is it true that ZG = ZH implies G 2: H? Many authors have worked on this question, but progress has been difficult [30]. Perhaps the best positive result was that of Whitcomb in 1968 [37], who showed that the implication G = H holds for G metabelian. Dade [9] showed there were counterexamples, even in the metabelian case, if Z were replaced by the family of all fields. Some mathematicians came to believe the problem was a kind of grammatical accident, that counterexamples for Z were surely there, if difficult to find. We ourselves began this investigation looking for counterexamples, but found that they were indeed very difficult to find, much more difficult than we had anticipated. Slowly we began to believe that at least some exciting positive results were possible. In this paper we answer the isomorphism problem for finite p-groups over the p-adic integers Z, = Zcp), and in a very strong way: In the normalized units (augmentation 1) of Z pG there is only one con&gay cZu.ss of groups of order ] G 1. This answers the isomorphism problem in the affirmative (for finite p-groups G, over Z or Zp) and in addition computes the entire Picard group [2] of the category of Z,G-modules in terms of automorphisms of G. Similarly we are able to settle the isomorphism problem for finite nilpotent groups and compute the associated semilocal Picard groups. We also treat more general coefficient domains: namely, integral domains S of characteristic 0 in which no (rational) prime divisor of the group order is invertible, for the SG isomorphism problem, and treat similar semilocal Dedekind domains for the Picard group computations. The Zassenhaus conjecture concerning the rational behavior of group ring automorphisms is verified for the nilpotent case in this general setting (cf. Corollary 3 below). Over Z, we announce a positive answer to the isomorphism

Group Rings and Class Groups

I Some general facts.- II Some notes on representation theory.- III The leading coefficient of units.- IV Class sum correspondence.- V More on the class sum correspondence.- VI Subgroup rigidity.- VII Global units.- VIII Locally isomorphic group rings.- IX Zassenhaus conjecture.- X Variations of the Zassenhaus conjecture.- XI Group Extensions.- XII Class sums of p-elements.- XIII Clifford theory revisited.- XIV Examples.- I Introduction and Review of the Tame Case.- II Hopf Orders.- III Principal Hornogeneous Spaces.- IV Arithmetic Applications:- The Cyclotomic Case.- V Arithmetic Applications:- The Elliptic Case.- References.

The isomorphism problem for integral group rings of finite groups

The modern theory of groups originated with the treatments of Galois (1811–1832), Cauchy (1789–1857) and Serret (1819–1885) on finite discontinuous substitution groups.

A Sylowlike theorem for integral group rings of finite solvable groups

By V(RG) we denote the units in RG, which have augmentation 1. The group of units in RG is then the product of the units in R and V(RG). A subgroup H of V(RG) with \H\ = |G| is called a group basis, provided the elements of H are linearly independent. This latter condition is automatic, provided no rational prime divisor of \H\ is a unit in R [1]. If H is a group basis, then RG = RH as augmented algebras and conversely. The object of this note is to prove the following

Outer group automorphisms may become inner in the integral group ring

Abstract In this note we shall construct a finite group G which has an automorphism α, which is not inner; however, the induced automorphism on SG is inner, where S is the ring of algebraic integers in a suitably chosen algebraic number field. A consequence of our arguments is that α is inner in KG for every field K .

Projective limits of group rings

A finite group G may be written as a projective limit of certain quotients Gi. Denote by Γ the corresponding projective limit of the integral group rings ZGi. The basic topic of the paper is the question whether Γ may be a replacement of ZG. In particular, this is studied in connection with the isomorphism problem of integral group rings and with the conjecture of Zassenhaus that different group bases of ZG are conjugate within QG. Using such projective limits, a Cech style cohomology set yields obstructions for these conjectures to be true, if G is soluble. This is used to construct two non-isomorphic groups as projective limits such that the projective limits of the corresponding group rings are semi-locally isomorphic. On the other hand, it is shown that for special classes of groups certain p-versions of the Zassenhaus conjecture hold. These p-versions are weaker than the conjecture but still provide a strong positive answer to the Isomorphism problem. In particular, such p-versions hold when G has a nilpotent commutator subgroup or when G is a Frobenius or a 2-Frobenius group.

GORENSTEIN TILED ORDERS

Let Λ = {O, E(Λ)} be a reduced tiled Gorenstein order with Jacobson radical R and J a two-sided ideal of Λ such that Λ ⊃ R 2 ⊃ J ⊃ Rn (n ≥ 2). The quotient ring Λ/J is quasi-Frobenius (QF) if and only if there exists p ∈ R 2 such that J = pΛ = Λp. We prove that an adjacency matrix of a quiver of a cyclic Gorenstein tiled order is a multiple of a double stochastic matrix. A requirement for a Gorenstein tiled order to be a cyclic order cannot be omitted. It is proved that a Cayley table of a finite group G is an exponent matrix of a reduced Gorenstein tiled order if and only if G = Gk = (2) × ⃛ × (2). Commutative Gorenstein rings appeared at first in the paper [3]. Torsion-free modules over commutative Gorenstein domains were investigated in [1]. Noncommutative Gorenstein orders were considered in [2] and [10]. Relations between Gorenstein orders and quasi-Frobenius rings were studied in [5]. Arbitrary tiled orders were considered in [4], 11-14.

Auslander-reiten quivers of schurian orders

We shall consider orders ⋀ over a complete discrete rank one valuation ring R in a split full matrix ring containing a complete sex of primitive orthogonal idempot ents. In case s of finite lattice type and R is the power series ring in one variable over its residexe class field k , we give a description of its index composable lattices and its Auslander-Reiten quiver in terms of representations of partially ordered sets. By a model theoretic argument, this implies a description of all indecomposable lattices if R is the completion of the ring of integers in an algebraic number field at all but possibly a finite number of primes.

Higher-dimensional orders, graph-orders, and derived equivalences ☆

Abstract Green-orders (tree-orders) in the classical one-dimensional case are the setting, to understand p -adic blocks with cyclic defect of finite groups. Blocks with “cyclic defect” of Hecke orders however, are Green-orders over two-dimensional rings. Hecke orders of dihedral groups of order divisible by 4 are even defined over a three-dimensional ring. We extend the notion of Green-orders to orders associated to a locally embedded graph instead of a tree, and to general complete regular local noetherian ground rings of finite dimension. We extend the result, that classical tree-orders are derived equivalent to star-orders. We then use these results to clarify the derived equivalence classes of tame algebras of Dihedral type.

Principal Homogeneous Spaces

As before, let K be the field of fractions of a Dedekind domain \( mathfrak{D} \) of characteristic 0, and let G be a finite abelian group. We will continue to work with Hopf orders \( mathfrak{A} \) in A = KG and B = Map(G,K). In this chapter we will be concerned with the objects on which a Hopf order in A acts. Rather than studying all \( mathfrak{A} \) -modules, we will make use of the comultiplication in \( mathfrak{A} \) by considering only those \( mathfrak{A} \) -modules which have the structure of an \( mathfrak{D} \) -algebra, and are in fact “twisted” versions of \( mathfrak{B} = \mathfrak{A}D \) . These objects are the principal homogeneous spaces for the Hopf order \( mathfrak{B} \) , and the set of isomorphism classes of principal homogeneous spaces can be given the structure of an abelian group \( PH(\mathfrak{B}) \) . As in the previous chapter, we will first work at the level of K-algebras, and then see how the theory lifts to integral level. We shall then construct a group homomorphism ψ from \( PH(\mathfrak{B}) \) to the locally free classgroup \( C1(\mathfrak{A}) \) . Finally, in the case that G is cyclic of order p and K contains a primitive pth root of unity, we use Kummer theory to give an explicit description of \( PH(\mathfrak{B}) \) and of the kernel of ψ.