On \(\pi\)-block induction in a \(\pi\)-separable group
Let \(\pi\) be a set of primes, and let \(G\) be a \(\pi\)-separable group. A nonempty set \(B\) of irreducible characters of \(G\) is called a \(\pi\)-block of \(G\) if it is a union of \(p\)-blocks for each \(p\in\pi\), and is minimal with respect to this property. Let \(D\) be a \(\pi\)-subgroup of \(G\), and let \(H\) be a subgroup of \(G\) such that \[ DC_G(D)\subseteq H\subseteq N_G(D).\tag{*} \] \textit{M. C. Slattery} [J. Algebra 124, No. 1, 236-269 (1989; Zbl 0684.20008)] defined a block induction map \(b\mapsto bi(H,G)\) from \(\pi\)-blocks of \(H\) to those of \(G\). Assuming always that (*) holds, the author proves two main results. First, for any \(\xi\in b\), there is \(\chi\in bi(H,G)\) with \([\chi,\xi^G]\neq 0\), and given \(\chi\in bi(H,G)\), there is \(\xi\in b\) such that \([\chi,\xi^G]\neq 0\). For the second result, assume that \(G\) is \(p\)-solvable, for some \(p\in\pi\), and that \(D\) has a \(p\)-subgroup \(P\) with \(C_G(P)\subseteq H\subseteq N_G(P)\). If \(b\) is a \(\pi\)-block of \(H\) and \(\mu\) is a \(p\)-block of \(H\) that is contained in \(b\), then \(\mu^G\) is contained in \(bi(H,G)\).
scientific article; zbMATH DE number 1563188