On Principal Blocks of p -Constrained Groups

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.

missing in Library copy.Introduction In recent years we have seen an increase in the number of group cohomology calculations. Starting with Leary’s work [L1] in the early nineties on the integral and mod-p cohomology rings of certain finite p-groups to more recent works on the cohomology of simple groups (see for example [AM1, AM2, AM3]), these calculations have not only expanded the collection of known cohomology rings but also introduced new techniques which have opened up a pathway for more work in this area. Despite these advances, group cohomology calculations remain generally difficult and much remains to be done towards obtaining amore coherent understanding of the cohomology of groups.In this thesis, we are mainly concerned with calculations of the cohomology rings of certain finite groups and the uses of some of these calculations in related topics. There are two main parts in this thesis with different techniques used in these parts.

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

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.

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.

Nilpotent decomposition in integral group rings

ABSTRACTLet G be a finite group and k an algebraically closed field of characteristic p. In this paper we investigate the Loewy structure of centers of indecomposable group algebras kG, for groups G with a normal elementary abelian Sylow p-subgroup. Furthermore, we show a reduction result for the case that a normal abelian Sylow p-subgroup is acted upon by a subgroup of its automorphism group; this is fundamental in providing generic formulae for the Loewy lengths considered.

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.

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.

It is shown that in the units of augmentation one of an integral group ring ℤ G of a finite group G, a noncyclic subgroup of order p 2, for some odd prime p, exists only if such a subgroup exists in G. The corresponding statement for p = 2 holds by the Brauer–Suzuki theorem, as recently observed by Kimmerle.

We study extensions of p-local finite groups where the kernel is a p-group. In particular, we construct examples of saturated fusion systems \({\mathcal{F}}\) which do not come from finite groups, but which have normal p-subgroups \({A \vartriangleleft \mathcal{F}}\) such that \({\mathcal{F}/A}\) is the fusion system of a finite group. One of the tools used to do this is the concept of a “transporter system”, which is modelled on the transporter category of a finite group, and is more general than a linking system.

A relation between algebraic degrees of eigenvalues of an element in the group ring Z G of a finite group G and their multiplicities is obtained. This result can be regarded as a theorem on asymptotic behavior of parameters in group rings of finite groups which is a parallel result of several well known theorems concerning the relationships of parameters in finite groups.

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.

Let G be a finite group and let w A IrrðGÞ lie in a p-block of full defect for an odd prime p. We investigate mQðwÞp, the p-part of the rational Schur index of w. At the onset of this work, it was suspected that these Schur indices would not be divisible by p, and in Section 2 below, we show this to be the case when G is either supersolvable or p-nilpotent. Further motivation came from the situation in which G has a normal Sylow p-subgroup. Of course in this case all irreducible characters of G lie in p-blocks of full defect. Recall that a p-hyperelementary subgroup is a subgroup with a cyclic, normal p-complement and that by results of Brauer, Witt, and Berman G has a p-hyperelementary subgroup H with z A IrrðHÞ such that mQðwÞp divides mQðzÞ. If the Sylow p-subgroup of G is normal, all p-hyperelementary subgroups of G are nilpotent. Hence, the Schur indices of their irreducible characters are at most 2, and it follows that p does not divide mQðwÞ: One final case is that in which the Sylow p-subgroups of G are abelian. In this situation, Brauer’s height zero conjecture implies that the degrees of the irreducible characters in p-blocks of full defect are not divisible by p. Thus their Schur indices are not divisible by p either. However, in Section 3 we fix an a A N and construct a solvable group G with irreducible character w lying in the principal p-block of G=KerðwÞ and satisfying mQðwÞ 1⁄4 p. It is crucial to regard w as a faithful irreducible character of G=KerðwÞ, as one can always regard an irreducible character c of a group H as an irreducible character of a larger group G which surjects onto H. This does not a¤ect the Schur index of c, but it can easily be set up to ensure c lies in the principal p-block of G: The group SLð2; 3Þ has a rational-valued irreducible character w such that