The Isomorphism Problem for Integral Group Rings of a Complete Monomial Group

Let p p be a prime. We denote by S n S_n the symmetric group of degree n n , by A n A_n the alternating group of degree n n and by F p {\mathbb F}_p the field with p p elements. An important concept of modular representation theory of a finite group G G is the notion of a block. The blocks are in one-to-one correspondence with block idempotents, which are the primitive central idempotents of the group ring F q G {\mathbb F}_q G , where q q is a prime power. Here, we describe a new method to compute the primitive central idempotents of F q G {\mathbb F}_q G for arbitrary prime powers q q and arbitrary finite groups G G . For the group rings F p S n {\mathbb F}_p S_n of the symmetric group, we show how to derive the primitive central idempotents of F p S n − p {\mathbb F}_p S_{n-p} from the idempotents of F p S n {\mathbb F}_p S_n . Improving the theorem of Osima for symmetric groups we exhibit a new subalgebra of F p S n {\mathbb F}_p S_n which contains the primitive central idempotents. The described results are most efficient for p = 2 p = 2 . In an appendix we display all primitive central idempotents of F 2 S n {\mathbb F}_2 S_n and F 4 A n {\mathbb F}_4 A_n for n ≤ 50 n \le 50 which we computed by this method.

Nilpotent decomposition in integral group rings

Weights for symmetric and general linear groups

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.

The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G) using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this thesis we introduce natural subspaces B(q,G)_p of B(q,G) defined for a fixed prime p. We show that B(q,G) is stably homotopy equivalent to a wedge sum of B(q,G)_p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial p-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial p-groups of rank at least 4 the space B(2,G) does not have the homotopy type of a K(π,1) space. Furthermore, we give a group theoretic condition, applicable to symmetric groups and general linear groups, which implies the space B(2,G) not having the homotopy type of a K(π,1) space. For a finite group G, we compute the complex K-theory of B(2,G) modulo torsion.

Clean group rings over localizations of rings of integers

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.

This paper studies the question, if $R$ is a ring satisfying a polynomial identity, what polynomial identities are satisfied by group rings and matrix rings over $R$? Theorem 2.6. If $R$ is an algebra over a field with at least $q$ elements, and $R$ satisfies ${x^q} = 0$, and $G$ is a group with an abelian subgroup of index $k$, then the group ring $R(G)$ satisfies ${x^t} = 0$, where $t = q{k^2} + 2$. Theorem 3.2. If $R$ is a ring satisfying a standard identity, and $G$ is a finite group, then $R(G)$ satisfies a standard identity. Theorem 3.4. If $R$ is an algebra over a field, and $R$ satisfies a standard identity, then the $k$-by-$k$ matrix ring ${R_k}$ satisfies a standard identity. Each theorem specifies the degree of the polynomial identity.

The mini workshop "Arithmetic of group rings" was attended by 16 participants from Belgium, Brazil, Canada, Germany, Hungary, Israel, Italy, Romania and Spain. The expertise was a good mixture between senior and young researchers. It was a very stimulating experience and thesize of the group allowed excellent discussions amongst all participants. Very fruitful were the problem sessions, resulting in the problems listed at the end of this report. The main highlights of the conference were: The complete calculation of the projective Schur subgroup of the Brauer group by Aljadeff and del Rio. Hertweck's solution of the first Zassenhaus conjecture for finite metacyclic groups. the description of special subgroups of the unit group of integral group rings, such as the hypercentre and the finite conjugacy centre, and the relation with respect to the normalizer of the trivial units. discussion of the present state of art via several survey talks presented and the problem sessions The group G determines its integral group ring \mathbb Z G and its group V(\mathbb Z G) of normalized units. Several talks addressed the interplay of the cohomological properties of these three objects. Further topics included twisted group rings, group rings over local rings, polynomial growth and identities, orders and semigroup rings, Lie structure, representation-theoretic and algorithmic methods.

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.

Let [Formula: see text] be the prime graph associated with a finite group [Formula: see text] and [Formula: see text] be the degree pattern of [Formula: see text]. A finite group [Formula: see text] is said to be [Formula: see text]-fold [Formula: see text]-characterizable if there exist exactly [Formula: see text] nonisomorphic groups [Formula: see text] such that [Formula: see text] and [Formula: see text]. The purpose of this paper is two-fold. First, it shows that the symmetric group [Formula: see text] is [Formula: see text]-fold [Formula: see text]-charaterizable. Second, it shows that there exist many infinite families of alternating and symmetric groups, [Formula: see text] and [Formula: see text], which are [Formula: see text]-fold [Formula: see text]-characterizable with [Formula: see text].

We use the theory of blocks of cyclic defect to prove that under a certain condition on the principal p p -block of a finite group G G , the normalized unit group of the integral group ring of G G contains an element of order p q pq if and only if so does G G for q q a prime different from p p . Using this we verify the Prime Graph Question for all alternating and symmetric groups and also for two sporadic simple groups.

Let G be a finite 2-generated non-cyclic group. The spread of G is the largest integer k such that for any nontrivial elements x1,…,xk, there exists y G ∈ G such that G = 〈xi,y〉 for all i. The more restrictive notion of uniform spread, denoted u(G), requires y to be chosen from a fixed conjugacy class of G, and a theorem of Breuer, Guralnick and Kantor states that u(G) ⩾ 2 for every non-abelian finite simple group G. For any group with u(G) ⩾ 1, we define the uniform domination number γu(G) of G to be the minimal size of a subset S of conjugate elements such that for each nontrivial x ∈ G there exists y ∈ S with G = 〈x, y〉 (in this situation, we say that S is a uniform dominating set for G). We introduced the latter notion in a recent paper, where we used probabilistic methods to determine close to best possible bounds on γu(G) for all simple groups G.In this paper we establish several new results on the spread, uniform spread and uniform domination number of finite groups and finite simple groups. For example, we make substantial progress towards a classification of the simple groups G with γu (G) = 2, and we study the associated probability that two randomly chosen conjugate elements form a uniform dominating set for G. We also establish new results concerning the 2-generation of soluble and symmetric groups, and we present several open problems.

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.

A BV-algebra structure on Hochschild cohomology of the integral group ring of finitely generated Abelian groups