Nilpotent characters
Let \(\pi\) be a set of primes, and let \(G\) be a finite \(\pi\)-separable group. Isaacs has defined a canonical basis \(I_\pi(G)\) for the vector space of class functions on the set of \(\pi\)-elements in \(G\). The author calls an element \(\varphi\in I_\pi(G)\) nilpotent if \(\varphi\) is induced from an element \(\delta\in I_\pi(K)\) where \(K\) is a subgroup of \(G\) containing a normal Hall \(\pi\)-subgroup. We can and will assume that \(\delta\) has \(\pi\)-degree, and we denote by \(B\) the \(\pi\)-block of \(G\) containing \(\varphi\). The author shows that \(\varphi\) is the only element of \(I_\pi(G)\) contained in \(B\) and that induction gives a bijection between \(\text{Irr}(K|\delta_{O_\pi(K)})\) and \(\text{Irr}(B)\); in particular, \(|\text{Irr}(B)|=|\text{Irr}(D)|\) where \(D\) denotes a defect group of \(B\). When \(\pi\) is the complement of a prime \(p\) then \(I_\pi(G)\) is the set \(\text{IBr}_p (G)\) of irreducible \(p\)-Brauer characters of \(G\), and an element \(\varphi\in\text{IBr}_p(G)\) is nilpotent if and only if the \(p\)-block \(B\) of \(G\) containing \(\varphi\) is nilpotent in the sense of Broué-Puig.
scientific article; zbMATH DE number 811720