the characteristic embedding of the Burnside ring

Let $G$ be a finite group, $H$ its subgroup and $X$ a finite $G$-set. By the $H$-fixed point subset of $X$ we understand the set

$$X^H=\{x\in X;{\forall }_{h\in H}hx=x\}.$$

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

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

$$|{(X\bigsqcup Y)}^H|=|X^H|+|Y^H|;$$

$$|{(X\times Y)}^H|=|X^H|\cdot |Y^H|.$$

Denote by $\mathrm{Sub}(G)=\{H\subseteq G;H\mathrm{is}\mathrm{a}\mathrm{subgroup}\mathrm{of}G\}$. Recall that any $H,K\in \mathrm{Sub}(G)$ are said to be conjugate iff there exists $g\in G$ such that $H=gK{g}^{-1}$. Conjugation is an equivalence relation. Denote by $\mathrm{Con}(G)$ the quotient set.

One can check that for any $H,K\in \mathrm{Sub}(G)$ such that $H$ is conjugate to $K$ and for any finite $G$-set $X$ we have

$$|X^H|=|X^K|.$$

Thus we have a well defined ring homomorphism:

$$\phi :\mathrm{\Omega }(G)\to \underset{(H)\in \mathrm{Con}(G)}{\oplus }\mathbb{Z};$$

$$\phi ([X]-[Y])={(|X^H|-|Y^H|)}_{(H)\in \mathrm{Con}(G)}.$$

This homomorphism is known as the characteristic embedding, since it is monomorphism (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	Derivation
Classification	msc 16S99