Computational aspects of Burnside rings, part I: the ring structure (original) (raw)
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Applications Of Burnside Rings In Elementary Group Theory
This is a report on some of the results which appear in DSY 90]. A canonical ring homomorphism from the Burnside ring (C) of a nite cyclic group C into the Burnside ring (G) of any nite group G of the same order is exhibited and it is shown that many results from elementary nite group theory, in particular those claiming certain congruence relations, are simple consequences of the existence of this map. Theorem: Let G be a nite group and let C denote the cyclic group of the same order n. There exists a ring homomorphism = (G) : (C) ?! (G) from the Burnside ring (C) of the cyclic group C into the Burnside ring (G) of the group G with the following property: for every subgroup U G of G and every element x 2 (C) one has ' U ((x)) = ' C jU j (x) where ' U ((x)) denotes the number of U{invariant elements in the virtual G{set (x) and C jUj denotes the unique subgroup of order jUj in C. Remark: This theorem gives a precise conceptual interpretation of the observation (Fr 95], Hu 67], Wa 80]) that quite a few elementary, but important results in group theory can be proved by comparing systematically certain group theoretic invariants of a nite group G with the same invariants for the cyclic group C of the same order. ' U (x d) = (d if d divides (G : U), 0 otherwise, In particular, U (x d) = 0 unless d divides (G : U) and U (x d) = (G : N G (U)) = #fgUg ?1 j g 2 Gg if (G : U) = d.
Modular Representations and Monomial Burnside Rings
2004
MODULAR REPRESENTATIONS AND MONOMIAL BURNSIDE RINGS Olcay Coskun M.S. in Mathematics Supervisors: Assoc. Prof. Dr. Laurence J. Barker and Asst. Prof. Dr. Ergun Yalcin August, 2004 We introduce canonical induction formulae for some character rings of a finite group, some of which follows from the formula for the complex character ring constructed by Boltje. The rings we will investigate are the ring of modular characters, the ring of characters over a number field, in particular, the field of real numbers and the ring of rational characters of a finite p−group. We also find the image of primitive idempotents of the algebra of the complex and modular character rings under the corresponding canonical induction formulae. The thesis also contains a summary of the theory of the canonical induction formula and a review of the induction theorems that are used to construct the formulae mentioned above.
2003
This thesis is concerned with some different aspects of the monomial Burnside rings, including an extensive, self contained introduction of the A−fibred G−sets, and the monomial Burnside rings. However, this work has two main subjects that are studied in chapters 6 and 7. There are certain important maps studied by Yoshida in [16] which are very helpful in understanding the structure of the Burnside rings and their unit groups. In chapter 6, we extend these maps to the monomial Burnside rings and find the images of the primitive idempotents of the monomial Burnside C−algebras. For two of these maps, the images of the primitive idempotents appear for the first time in this work. In chapter 7, developing a line of research persued by Dress [9], Boltje [6], Barker [1], we study the prime ideals of monomial Burnside rings, and the primitive idempotents of monomial Burnside algebras. The new results include; (a): If A is a π−group, then the primitive idempotents of Z (π) B(A, G) and Z (π) B(G) are the same (b): If G is a π −group, then the primitive idempotents of Z (π) B(A, G) and QB(A, G) are the same (c): If G is a nilpotent group, then there is a bijection between the primitive idempotents of Z (π) B(A, G) and the primitive idempotents of QB(A, K) where K is the unique Hall π −subgroup of G. (Z (π) = {a/b ∈ Q : b / ∈ ∪ p∈π pZ}, π =a set of prime numbers).
Abstracts from the 20th Conference on Applications of Computer Algebra, ACA 2014
2014
This is the first edition of a special session at ACA conference devoted to providing a forum for exchange of ideas and research results related to Computer Algebra Aspects of Finite Rings in a broad sense and also to their applications. Session topics will include (but are not limited to) the following: 1) Computer Algebra and Finite Rings Computer representation and computation over finite rings and polynomial rings over finite rings. C.A. software and development of packages devoted to finite rings and related structures ...
Congruences for the Burnside module
Hokkaido Mathematical Journal, 2003
Let G be a finite group. Oliver-Petrie constructed a \Pi-complex for a finite G-CW-complex and defined a Burnside module \Omega(G, \Pi) which consists of equivalent classes of all \Pi-complexes. It is well-known that a congruence holds for the Burnside ring. The purpose of this paper is to prove congruences for the Burnside module.
2021
This paper develops links between the Burnside ring of a finite group G and the slice Burnside ring. The goal is to gain a better understanding of ghost maps, idempotents, prime spectrum of these Burnside rings and connections between them. MSC(2010): 19A22, 18G30, 06A11, 20J15