Blocks relative to a normal subgroup in \(\pi\)-separable groups
Let \(\pi\) be a set of primes, let \(N\) be a normal subgroup of a finite group \(G\) such that \(G/N\) is \(\pi\)-separable , and let \(H/N\) be a Hall \(\pi\)-subgroup of \(G/N\). The author shows that the vector space of complex class functions on \(G\) vanishing on \(G-\bigcup_{g\in G}H^g\) has a unique basis \(P(G|H)\) satisfying the following properties: (I) For \(\beta\in P(G|H)\), there is \(\gamma\in\text{Irr}(H)\) such that \(\beta=\gamma^G\); (D) For \(\alpha\in\text{Irr}(H)\), \(\alpha^G\) is a nonnegative integer combination of elements of \(P(G|H)\). Then \(\chi,\psi\in\text{Irr}(G)\) are called \(N\)-\(\pi\)-linked if \([\chi,\eta][\psi,\eta]\neq 0\) for some \(\eta\in P(G|H)\). This defines a graph \(\Gamma\) whose vertices are the irreducible characters of \(G\). The connected components of \(\Gamma\) are called the \(N\)-\(\pi\)-blocks of \(G\). Let \(B\) be an \(N\)-\(\pi\)-block of \(G\), and let \(L/N\) be a normal \(\pi\)-subgroup of \(G/N\). The author shows: (i) There exists \(\vartheta\in\text{Irr}(L)\) such that \(B\subseteq\text{Irr}(G|\vartheta)\); (ii) There exists an \(N\)-\(\pi\)-block \(b\) of the stabilizer \(T\) of \(\vartheta\) in \(G\) such that \(B=\{\psi^G:\psi\in b\}\); (iii) If \(L/N=O_\pi(G/N)\) and if \(\vartheta\) is \(G\)-stable then \(B=\text{Irr}(G|\vartheta)\); (iv) If \(\alpha\in\text{Irr}(H)\) is quasiprimitive then \(\alpha^G\in P(G|H)\).
scientific article; zbMATH DE number 1539393