On the representation theory of \(\pi\)-separable groups
The paper under review gives a new and simplified approach to results of \textit{I. M. Isaacs} [J. Algebra 86, 98-128 (1984; Zbl 0526.20006)] on characters of \(\pi\)-separable groups where \(\pi\) is a set of primes. More specifically, the authors prove that, for any \(\pi\)-separable group, the group of generalized characters vanishing on \(\pi\)-singular elements has a basis such that every character vanishing on \(\pi\)-singular elements can be written as a non-negative linear combination of basis elements. They also show that the group of generalized characters restricted to \(\pi\)-regular elements has a basis such that the restriction of every character to \(\pi\)-regular elements can be written as a non-negative linear combination of basis elements. This generalizes results of P. Fong for the case where \(\pi\) consists of a single prime. In the proof the authors make use of the representation theory of finite groups over certain semilocal rings.
scientific article; zbMATH DE number 4071180