Numbers of classes and chains of subgroups in finite groups
Let \(X\) be a normal subset of a finite group \(G\), and let \(m_N(V)\) denote the number of conjugacy classes of \(N\) contained in \(V\cap X\), for subgroups \(V\subseteq N\subseteq N_G(V)\subseteq G\). Moreover, let \(\mathcal F\) be a conjugation-invariant collection of subgroups of \(G\) such that each element \(x\in X\) is contained in a unique smallest member \(F_x\in{\mathcal F}\). Finally, let \(\mathcal S\) denote the \(G\)-poset of nonempty chains in \(\mathcal F\), and let \(G_\sigma\) denote the stabilizer and \(V_\sigma\) the smallest member of a chain \(\sigma\in{\mathcal S}\). The author shows that \(m_G(G)=\sum_{\sigma\in{\mathcal S}}(-1)^{|\sigma|+1} m_{G_\sigma}(V_\sigma)/| G:G_\sigma|\). This, together with a related theorem, generalizes earlier results by \textit{G. Robinson} and this reviewer [e.g., J. Algebra 205, No. 1, 326-332 (1998)]. The proofs are elementary and simplify the earlier representation-theoretic approach.
scientific article; zbMATH DE number 1213844