### On the Burnside Algebra of a Finite Group

#### 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 ¡æ 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).

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