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On a character correspondence associated to $\mathfrak{F}$-projectors

María José Felipe, Iris Gilabert, Lucia Sanus

TL;DR

This work studies head characters of finite solvable groups, originally tied to Carter subgroups, and analyzes their restriction behavior and kernel intersections. It extends Isaacs' results to a broad framework of saturated formations $\mathfrak{F}$ containing the nilpotent formation via Navarro's $\mathfrak{F}'$-characters, introducing strong $H$-pair series as a central tool. A descending characterization of $\mathfrak{F}'$-characters is developed, yielding a counting formula $|\mathrm{Irr}_{\mathfrak{F}'}(G)|=|\mathrm{Irr}(H/H')|$ and enabling generalized A- and B-type theorems: restriction behavior of $\mathfrak{F}'$-characters and the structure of their kernels in relation to $\mathfrak{F}$-projectors. These results unify and extend classical results for Carter subgroups and $p'$-degree characters (McKay-type results) to a broad, formation-theoretic setting, with implications for character correspondences in solvable groups.

Abstract

We study the conditions under which the head characters of a finite solvable group, as defined by I. M. Isaacs, behave well with respect to restriction. We also determine the intersection of the kernels of all head characters of the group. Using G. Navarro's definition of $\mathfrak{F}'$-characters, we generalize these results for any saturated formation $\mathfrak{F}$ containing the formation of nilpotent groups.

On a character correspondence associated to $\mathfrak{F}$-projectors

TL;DR

This work studies head characters of finite solvable groups, originally tied to Carter subgroups, and analyzes their restriction behavior and kernel intersections. It extends Isaacs' results to a broad framework of saturated formations containing the nilpotent formation via Navarro's -characters, introducing strong -pair series as a central tool. A descending characterization of -characters is developed, yielding a counting formula and enabling generalized A- and B-type theorems: restriction behavior of -characters and the structure of their kernels in relation to -projectors. These results unify and extend classical results for Carter subgroups and -degree characters (McKay-type results) to a broad, formation-theoretic setting, with implications for character correspondences in solvable groups.

Abstract

We study the conditions under which the head characters of a finite solvable group, as defined by I. M. Isaacs, behave well with respect to restriction. We also determine the intersection of the kernels of all head characters of the group. Using G. Navarro's definition of -characters, we generalize these results for any saturated formation containing the formation of nilpotent groups.

Paper Structure

This paper contains 8 sections, 27 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Some properties on projectors
  3. Some properties on projectors
  4. Extensions of irreducible characters
  5. $\mathfrak{F}'$-characters
  6. Strong pair series
  7. Theorem A
  8. Theorem B

Key Result

Lemma 2.1

Let $\mathfrak{F}$ be a saturated formation and $G$ a solvable group. There exists exactly one conjugacy class of subgroups $H$ of $G$ such that, for every normal subgroup $N$ of $G$, the subgroup $HN/N$ satisfies the following properties: In other words, $HN/N$ is $\mathfrak{F}$-maximal in $G/N$ for all normal subgroups $N$ of $G$.

Theorems & Definitions (49)

  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3: Theorem IV.5.18 of Doerk
  • Remark 1
  • Lemma 2.4
  • Lemma 3.1: Corollary 4.3 of Ispi
  • Lemma 3.2: Corollary 4.2 of Ispi
  • Theorem 3.3: Theorem 3.5 of N
  • Theorem 3.4
  • ...and 39 more