We start with a short historical background. The General Burnside problem (original) (raw)
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PI-groups and PI-representations of groups
Journal of Mathematical Sciences, 2009
It is well known that many famous Burnside-type problems have positive solutions for P I-groups and P I-algebras. In the present article we also consider various Burnside-type problems for P I-groups and P Irepresentations of groups. Contents 1 Introduction 1 2 Algebraic P I-groups 4 3 Unipotent P I-representations of groups 5
On pi\ pi pi-extensions of the semigroup mathbbZ+\ mathbb {Z} _+ mathbbZ+
2013
Abstract: We study inverse pi\ pi pi-extensions of the semigroup mathbbZ+\ mathbb {Z} _+ mathbbZ+. It is shown that pi\ pi pi-extension of the semigroup mathbbZ+\ mathbb {Z} _+ mathbbZ+ is inverse, iff its pi\ pi pi-extension coincides with pi(mathbbZ+)\ pi (\ mathbb {Z} _+) pi(mathbbZ+). The existence of a non-inverse pi\ pi pi-extension for semigroup mathbbZ+\ mathbb {Z} _+ mathbbZ+ is proved.
On the product of two $ pi$-decomposable groups
Revista Matematica Iberoamericana, 2014
The aim of this paper is to prove the following result: Let π be a set of odd primes. If the finite group G = AB is a product of two πdecomposable subgroups A = Oπ(A) × O π (A) and B = Oπ(B) × O π (B), then Oπ(A)Oπ(B) = Oπ(B)Oπ(A) and this is a Hall π-subgroup of G.
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).
Math Comput, 2008
We present an alternative constructive proof of the Brauer-Witt theorem using the so-called strongly monomial characters that gives rise to an algorithm for computing the Wedderburn decomposition of semisimple group algebras of finite groups.
Twisted Burnside theorem for type II${}_1$ groups: an example
Mathematical Research Letters, 2006
The purpose of the present paper is to discuss the following conjecture of Fel'shtyn and Hill, which is a generalization of the classical Burnside theorem: Let G be a countable discrete group, φ its automorphism, R(φ) the number of φconjugacy classes (Reidemeister number), S(φ) = # Fix(b φ) the number of φ-invariant equivalence classes of irreducible unitary representations. If one of R(φ) and S(φ) is finite, then it is equal to the other. This conjecture plays a very important role in the theory of twisted conjugacy classes having a long history (see [14], [4]) and has very serious consequences in Dynamics, while its proof needs rather fine results from Functional and Non-commutative Harmonic Analysis. It was proved for finitely generated groups of type I in [10]. In the present paper this conjecture is disproved for non-type I groups. More precisely, an example of a group and its automorphism is constructed such that the number of fixed irreducible representations is greater than the Reidemeister number. But the number of fixed finite-dimensional representations (i.e. the number of invariant finite-dimensional characters) in this example coincides with the Reidemeister number. The directions for search of an appropriate formulation are indicated (another definition of the dual object).
Congruences on bands of \b{π}-groups
Discussiones Mathematicae - General Algebra and Applications, 2013
A semigroup S is said to be completely π-regular if for any a ∈ S there exists a positive integer n such that a n is completely regular. The present paper is devoted to the study of completely regular semigroup congruences on bands of π-groups.