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On the Burnside Algebra of a Finite Group
Dilip P. Patil,Anshoo Tandon +1 more
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In this article, it was shown that the canonical Burnside mark homomorphism of the Burnside algebra B(G) of a finite group G into the product Z-algebra of rank #CG is injective.Abstract:
In this article first we shall prove the classical theorem of Burnside which asserts that the canonical Burnside mark homomorphism of the Burnside algebra B(G) of a finite group G into the product Z-algebra of rank #CG is injective, where CG denote the set of conjugacy classes of the subgroups of G. We further prove that for any finite group G the canonical Z-algebra homomorphism ZCZG iae ZCG maps the Burnside algebra B(ZCG G ) of a finite cyclic group ZG of order #G into the Burnside algebra B(G). We deduce quite a few elementary, but important results in finite group theory by using this canonical algebra homomorphism. Finally we describe the prime spectrum SpecB(G) and maximal spectrum SpmB(G) of B(G).read more
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References
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Book
Theory of Groups of Finite Order
TL;DR: In this article, the authors define the notion of permutation groups as a group of linear substitutions, and show that a group can be represented as a permutation-group.
Journal ArticleDOI
A characterisation of solvable groups
TL;DR: In this paper, it was shown that G is solvable if and only if the prime ideal spectrum Spec(~2(G)) of Q(G) is connected in the Zariski topology.
Journal ArticleDOI
The Burnside algebra of a finite group
TL;DR: The Burnside ring of G is a semisimpleteness algebra over Q and formulas for certain primitive idempotents of this algebra yield the theorem of Artin on rational characters.
Journal ArticleDOI
An application of Burnside rings in elementary finite group theory
TL;DR: A canonical map from the Burnside ring Ω(C) of a finite cyclic group C into the burnside ring of any finite group G of the same order is presented in this article.
MonographDOI
Introduction to algebraic geometry and commutative algebra
Dilip P Patil,Uwe Storch +1 more
TL;DR: Finitely Generated Algebras The K-Spectrum and the Zariski Topology Prime Spectra and Dimension Schemes Projective Scheme Regular, Normal and Smooth Points Riemann-Roch Theorem as mentioned in this paper