On local control of the number of conjugacy classes (Q1266469)

In this short technical paper everything is focussed around the proof of one result, namely Theorem: Let \(G\) be a finite group. Then \[ k^*(G)=\sum_{\sigma\in{\mathcal S}^*(G)/G}(-1)^{|\sigma|+1}k^*_{G_\sigma}(V_\sigma) \] (here is the notion of the symbols: \({\mathcal S}^*(G)=\) set of non-empty chains in \({\mathcal S}(G)\), \({\mathcal S}(G)=\) the simplicial complex associated to the poset of non-trivial solvable subgroups of \(G\), \(k^*_X(Y)=\) number of \(X\)-conjugacy classes of \(Y^*\), where the group \(X\) acts by conjugation on the group \(Y\), \(Y^*=Y\setminus\{1\}\), \(V_\sigma=\) initial subgroup of the chain \(\sigma\in{\mathcal S}(G)\), \(|\sigma|=\) number of non-trivial subgroups in the simplex \(\sigma\in{\mathcal S}(G)\)).