Primitive Characters and Permutation Characters of Solvable Groups

TL;DR

This paper explores the relationship between primitive characters and permutation characters in finite solvable groups, establishing a unique association with certain subgroups and applying this to character properties.

Contribution

It introduces a novel method to associate primitive characters with specific subgroups via permutation characters in solvable groups.

Findings

Unique association between primitive characters and subgroup conjugacy classes

Characterization of primitive characters through permutation characters

Applications to properties of primitive characters in solvable groups

Abstract

Let X be an irreducible, primitive complex character of the finite solvable group G, and let X* denote the complex conjugate character. If the degree X(1) is odd, then we show how to associate to X in a unique way, a conjugacy class of subgroups U of G for which X*X = (1_U)^G, the permutation character on the cosets of U. We investigate this situation and give a number of applications to properties of primitive characters of solvable and p-solvable groups.