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

XH={xX;hHhx=x}.

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

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

|(XY)H|=|XH|+|YH|;
|(X×Y)H|=|XH||YH|.

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

|XH|=|XK|.

Thus we have a well defined ring homomorphismMathworldPlanetmath:

φ:Ω(G)(H)Con(G);
φ([X]-[Y])=(|XH|-|YH|)(H)Con(G).

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

References

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