Generalized permutation characters of solvable groups
Let \(\chi\) be an irreducible complex character of the finite group G and \(\sigma\in Aut(G)\). Write \(sp(\chi)\) for the sum of the distinct Galois conjugates of \(\chi\) ; here a Galois conjugate \(\chi^{\sigma}\) is the irreducible character of G determined by \((\chi^{\sigma})(g) = (\chi(g))^{\sigma}\). Define P(G) as the set of integral linear combinations of ordinary permutation characters of G. The following is a theorem by E. Artin: Let \(\chi\in Irr(G)\). Then \(| G| \cdot sp(\chi)\in P(G)\). Therefore there is a least positive integer \(n(\chi)\) such that \(n(\chi)sp(\chi)\in P(G).\) The main bulk of this paper is devoted to the proof of the following theorems. (A) Let G be solvable and \(\chi\in Irr(G)\). Then the odd part of \(n(\chi)\) divides \(\chi(1).\) (B) Let G be solvable and \(\chi\in Irr(G)\). Assume \(\chi\) (1) is a power of 2. Then n(\(\chi)\) divides \(\chi(1).\) [In the author's paper in J. Algebra 93, 439-444 (1985; Zbl 0594.20006), it was shown that, for every \(2^ a\), \(a\geq 1\), there is a solvable group G and some \(\chi\in Irr(G)\) such that \(\chi\) (1) is odd and \(2^ a\) divides \(n(\chi)\).] The paper under review contains several substantial results on supersolvable groups, symplectic modules, cyclic normal subgroups, canonical characters, permutation characters, which makes it an important basic one to study.
scientific article; zbMATH DE number 3959742