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Above Isaacs' Head Characters

Asier Arranz

TL;DR

The paper extends the inductive McKay framework to Isaacs' head characters for a normal solvable subgroup $N$ of a finite group $G$, establishing a precise equivariant link between $\mathrm{H}(N)$ and $\mathrm{Lin}(C)$ for a Carter subgroup $C$ of $N$ via isomorphic character triples. It builds a robust apparatus of strong character-triple isomorphisms and inductive Ext-bijections (centered on Carter subgroups, intermediate subgroups, and orbit considerations) to compare $G$-character theory with $\mathbf{N}_G(C)$. A key consequence is the lower bound $k(G)\ge k(\mathbf{N}_G(C)/C')$ with equality characterizing when $N$ is abelian; the results specialize to known results for solvable $N$, self-normalizing Sylow subgroups, and $p'$-characters, and connect to Navarro’s canonical bijection. Overall, the work unifies McKay-type correspondences and head-character theory in an automorphism- and Galois-equivariant framework, yielding practical bounds on conjugacy classes and canonical bijections in the presence of Carter subgroups.

Abstract

The proof of the inductive McKay condition has been shown to imply that the character theory above the characters of degree not divisible by $p$ of a normal subgroup is locally determined. In this note, we establish a similar result for the Isaacs' head characters of a normal solvable subgroup of an arbitrary group. In particular, we give a new lower bound of the number of conjugacy classes of a finite group in terms of the Carter subgroups of any of its normal solvable subgroups.

Above Isaacs' Head Characters

TL;DR

The paper extends the inductive McKay framework to Isaacs' head characters for a normal solvable subgroup of a finite group , establishing a precise equivariant link between and for a Carter subgroup of via isomorphic character triples. It builds a robust apparatus of strong character-triple isomorphisms and inductive Ext-bijections (centered on Carter subgroups, intermediate subgroups, and orbit considerations) to compare -character theory with . A key consequence is the lower bound with equality characterizing when is abelian; the results specialize to known results for solvable , self-normalizing Sylow subgroups, and -characters, and connect to Navarro’s canonical bijection. Overall, the work unifies McKay-type correspondences and head-character theory in an automorphism- and Galois-equivariant framework, yielding practical bounds on conjugacy classes and canonical bijections in the presence of Carter subgroups.

Abstract

The proof of the inductive McKay condition has been shown to imply that the character theory above the characters of degree not divisible by of a normal subgroup is locally determined. In this note, we establish a similar result for the Isaacs' head characters of a normal solvable subgroup of an arbitrary group. In particular, we give a new lower bound of the number of conjugacy classes of a finite group in terms of the Carter subgroups of any of its normal solvable subgroups.
Paper Structure (4 sections, 9 theorems, 15 equations)

This paper contains 4 sections, 9 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. A Strong Character-Triple Isomorphism
  3. The Inductive Step
  4. Proof of Theorem \ref{['thmA:A']}

Key Result

Theorem 2.1

Assume that $K/L$ is a strong section of $G$ and let $K\subseteq N\lhd G$ such that $|K:L|$ and $|N:K|$ are relatively prime. Let $\theta\in \textnormal{Irr}(K)$ and $\varphi\in \textnormal{Irr}(L)$ be fully ramified with respect to $K/L$, where $\varphi$ is $G$-invariant. Then there exist a complem such that the following hold.

Theorems & Definitions (9)

  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 4.3