The canonical character of metacyclic groups with a form-quasiprimitive symplectic module and an application to Brauer formulas of local Galois groups (Q919091)

In the paper the canonical character (in the sense of \textit{I. M. Isaacs} [Am. J. Math. 95, 594-635 (1973; Zbl 0277.20008)]) of metacyclic groups with a faithful, irreducible, form-quasiprimitive, symplectic \({\mathbb{F}}_ pG\)-module is decomposed into a sum of characters induced from Abelian characters. Moreover the problem of finding explicit Brauer formulas (resulting from Brauer's induction theorem) for local Galois groups of p- adic number fields is reduced to the problem of finding a Brauer formula for the canonical character of tame local Galois groups. These groups are metacyclic. Thus many of such groups can be handled with the first mentioned result.