the characteristic embedding of the Burnside ring

Let G be a finite groupMathworldPlanetmath, H its subgroupMathworldPlanetmathPlanetmath and X a finite G-set. By the H-fixed pointPlanetmathPlanetmath subset of X we understand the set


Denote by |X| the cardinality of a set X.

It is easy to see that for any G-sets X,Y we have:


Denote by Sub(G)={HG;HisasubgroupofG}. Recall that any H,KSub(G) are said to be conjugate iff there exists gG such that H=gKg-1. ConjugationMathworldPlanetmath is an equivalence relationMathworldPlanetmath. Denote by Con(G) the quotient set.

One can check that for any H,KSub(G) such that H is conjugate to K and for any finite G-set X we have


Thus we have a well defined ring homomorphismMathworldPlanetmath:


This homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is known as the characteristic embedding, since it is monomorphismMathworldPlanetmathPlanetmathPlanetmath (see [1] for proof).


Title the characteristic embedding of the Burnside ring
Canonical name TheCharacteristicEmbeddingOfTheBurnsideRing
Date of creation 2013-03-22 18:08:09
Last modified on 2013-03-22 18:08:09
Owner joking (16130)
Last modified by joking (16130)
Numerical id 7
Author joking (16130)
Entry type DerivationPlanetmathPlanetmath
Classification msc 16S99