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
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).