Fong characters and chains of normal subgroups. (Q2458398)

Let \(\pi\) be a set of primes, \(G\) a \(\pi\)-separable finite group, and \(H\) a Hall \(\pi\)-subgroup. \(\alpha\in\text{Irr}(H)\) is said to be `Fong' for a \(\pi\)-partial \(\varphi\in\text{Irr}(G)\) if \(\alpha\) is a constituent of \(\varphi_H\) of least degree, and \(\alpha\in\text{Irr}(H)\) is said to be `\(G\)-Fong' if there is some \(\pi\)-partial \(\varphi\in\text{Irr}(G)\) such that \(\alpha\) is Fong for \(\varphi\). Isaacs asked whether for \(\pi\)-partial \(\varphi\in\text{Irr}(G)\) there always is an \(\alpha\in\text{Irr}(H)\) such that \(\alpha\) is Fong for \(\varphi\) and every irreducible constituent of \(\alpha_{H\cap N}\) is \(N\)-Fong for every normal subgroup \(N\) of \(G\). In general, the question is still open, and in the paper under review the authors prove some results related to this question. For example, they prove that the answer to Isaacs' question is yes if the normal subgroups of \(G\) are totally ordered by inclusion. The answer is also yes if \(N_G(H)\) acts transitively on the Fong characters associated with \(\varphi\). Moreover, it is shown that if \(\alpha\in\text{Irr}(H)\) is quasiprimitive, then the irreducible constituents of \(\alpha_{H\cap N}\) are \(N\)-Fong for every normal subgroup \(N\) of \(G\).