Groups with abundant Fong characters (Q1333209)

In [J. Algebra 99, 89-107 (1986; Zbl 0587.20005)] \textit{I. M. Isaacs} proves an interesting characterization of Fong characters among the irreducible characters of a Hall subgroup. At the end of his paper he observes that if \(G\) is a \(\pi\)-separable group with \(\pi\)-length 1, then \(G\) satisfies the following: Property \(P(\pi)\). Every irreducible character of a Hall \(\pi\)-subgroup is a Fong character. Isaacs describes an example of a group with \(\pi\)-length 2 which has this property, and in [loc. cit., Question 9.5] he asks how large the \(\pi\)- length of a group with Property \(P(\pi)\) can be; our aim is to show that for suitable choices of \(\pi\) it can be as large as you like. In [the author, J. Algebra 167, No. 3, 557-577 (1994; Zbl 0808.20009)] a class \({\mathfrak X}=\bigcup^\infty_{r=1} {\mathfrak X} (r)\) of soluble groups is constructed to answer a certain question about the influence of the group algebra on the structure of a group. We shall prove that groups in \(\mathfrak X\) satisfy the conclusions of the following Theorem: Let \(\pi\) be a set of primes, let \(G\) be a group in the class \(\mathfrak X\) and let \(H\) be a Hall \(\pi\)-subgroup of \(G\). Let \(\chi \in B_\pi (G)\) and write \(\chi_H=\psi_1 + \psi_2 + \cdots + \psi_r\) with \(\psi_i \in \text{Irr} (H)\) for \(i=1, \dots, r\). Then (a) \(\psi_1(1)=\psi_2(1)=\cdots=\psi_r(1)\) and (b) for each \(i=1, 2, \dots, r\) the index \(|H : \text{Ker} (\psi_i)|\) is divisible by the largest prime in \(\pi\) that divides \(|G/ \text{Ker} (\chi)|.\)