Finite dimensional group rings

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.

Terms like commutativity degree, non-commuting graph and isoclinism are far well-known for much of the group theorists nowadays. There are so many papers about each of these concepts and also about their relationships in finite groups. Also, there are some recent researches about generalizing these notions in finite rings and their connexions.The concepts of commutativity degree and non-commuting graph are also extended to non-associative structures such as Moufang loops and some part of the known results in group theory in these contexts have been expanded to them.In this paper, we are going to generalize the notion of isoclinism in finite Moufang loops and then study the relationships between these three concepts. Among other results, we prove that two isoclinic finite Moufang loops have the same commutativity degree and if they have the same sizes of centers and commutants then they have isomorphic non-commuting graphs. Also, the converses of these results have been investigated.Furthermore, it has been proved that a finite simple group can be characterized by its non-commuting graph. We will prove the same is true for a finite simple Moufang loop by imposing one additional hypothesis, namely, the isoclinism of the regarding loops.

Nilpotent decomposition in integral group rings

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.

Modular invariance property of association schemes is recalled in connection with our joint work with François Jaeger. Then we survey codes over F 2 discussing how codes, through their (various kinds of) weight enumerators, are related to (various kinds of) modular forms through polynomial invariants of certain finite group actions and theta series. Recently, not only codes over an arbitrary finite field but also codes over finite rings and finite abelian groups are considered and have been studied extensively. We show how the determination of the solutions of the modular invariance property of finite abelian groups (our joint work with Jaeger) is used to define the concept of Type II codes over arbitrary finite abelian groups. As an example of the usefulness of such Type II codes, we give an application to hermitian modular forms.

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.

The purpose of this paper is an extension of a theorem of Hughes (1973). He showed: Let R be a ring which has a right artinian right quotient ring and let G be a group which has a (transfinite) ascending normal series with each factor either finite or cyclic, but only a finite number of finite factors. Then the group ring RG has a right artinian right quotient ring.

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 R be a non trivial finite commutative ring with identity and G be a non trivial group. We denote by P(RG) the probability that the product of two randomly chosen elements of a finite group ring RG is zero. We show that P(RG) <0.25 if and only if RG is not isomorphic to Z2C2, Z3C2, Z2C3. Furthermore, we give the upper bound and lower bound for P(RG). In particular, we present the general formula for P(RG), where R is a finite field of characteristic p and |G| ≤ 4.

Expanders are highly connected sparse graphs of great interest in computer science, in areas ranging from parallel computation to complexity theory, from cryptography to coding theory, and, most recently, computational group theory (see e.g. [AKS, G+, LP, SS, V, WZ].) The explicit constructions of expander graphs [M1, M2, LPS] (see also [GG, Lu]) use deep mathematical tools to construct families of Cayley graphs of finite groups. The Independence Problem [LW], is whether being an expander family is a property of the groups alone, independent of the choice of generators. A counterexample to the general the problem was recently obtained in [ALW], by using a new combinatorial construction of expanders [RVW]. The problem remains for all classical series of finite simple groups (An, PSL(2,p), etc.)Let G be a finite group generated by at most d elements. The product replacement graph Γk(G) is defined to be a graph, with vertices corresponding to generating k-tuples in G, and edges corresponding to Nielsen transformations. Most recently, graphs Γk(G) became a subject of an intense investigation, prompted by the study of a commonly used 'practical' product replacement algorithm for generating random elements in finite groups, designed Leedham-Green and Soicher [LG]. This algorithm, based on the random walk on graphs Γk(G), showed a remarkable performance, as reported in [C+]. It was suggested in [LP], and proved in several special cases, that the product replacement graphs Γk(G) are expanders, for a fixed k, when |G| → ∞The main result of this paper is Theorem 2, establishing the connection between the expansion coefficient of the product replacement graph Γk(G) and the minimal expansion coefficient of a Cayley graph of group G with k generators. One can think of our result as of an additional evidence in favor of the speculation in [LP]. On the other hand, it gives an algorithmic motivation for study of the Independence Problem, in the aftermath of [ALW].In a special case (see Theorem 1 below), we show that if one assumes that all Cayley graphs with at most four generators in PSL (2, p) have a universal lower bound on expansion, then the product replacement graphs Γk(PSL(2,p)) form an expander family, when k ≥ 8 is fixed, and p → ∞.The proof of the Theorems consists of two parts: probability on groups and graph theoretic. The latter is based on a new combinatorial result of independent interest (Lemmas 4, 5), which generalizes a classical lemma that the direct product of expanders is an expander (see e.g. [Ch]). We prove that if a graph Γ on a direct product of sets satisfies two conditions:1) all 'row' subgraphs are expanders,2) at least (1-e) proportion of 'column' graphs are expanders,then graph &Gamma is also an expander. We use a novel combinatorial technique based on graph decomposition, as opposed to path arguments used in previously in [CG, DS2, DS3, P4]. It is easy to see that such a technique can never prove that a certain family of graphs is an expander family (cf. [P3]).

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:

We study certain properties of finite groups in which each subgroup of the commutant is invariant. Finite nonnilpotent groups with this property are completely described.