A remark on Brauer's k(B)-conjecture
``Forgetting about context, one could say that this paper does just one thing, namely, it constructs the square root of a generalized character.'' After this unusual opening remark the author describes the generalized character \(\delta\) of a finite group G having a ``square root'': Let G act on a finite set \(\Omega\) and let F be a finite field of characteristic prime to \(| \Omega | =n\). If \(g\in G\) has m orbits on \(\Omega\), then \(\delta (g)=| F|^{n-m}\). The motivation for the result lies in the author's significant contributions to R. Brauer's k(B)-conjecture. Its corollary is: Let the p\({}'\)-group G act faithfully on the elementary abelian p-group V. Suppose that there is a \(v\in V\) so that the restriction of V to \(C_ G(V)\) is a permutation module. Then the semidirect product GV has at most \(| V|\) conjugacy classes.
scientific article; zbMATH DE number 4165157