# 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:

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

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 $H=gKg^{-1}$. 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:

 $\varphi:\Omega(G)\rightarrow\bigoplus_{(H)\in\mathrm{Con}(G)}\mathbb{Z};$
 $\varphi([X]-[Y])=(|X^{H}|-|Y^{H}|)_{(H)\in\mathrm{Con}(G)}.$

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

## References

• 1 T. tom Dieck, , Lecture Notes in Math. 766, Springer-Verlag, Berlin, 1979.
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