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A reduction theorem for the Character Triple Conjecture

Damiano Rossi

Abstract

In this paper, we show that the Character Triple Conjecture holds for all finite groups once assumed for all quasi-simple groups. This answers the question on the existence of a self-reducing form of Dade's conjecture, a problem that was long investigated by Dade in the 1990s. Our result shows that this role is played by the Character Triple Conjecture, recently introduced by Späth, that we present here in a general form free of all previously imposed restrictions.

A reduction theorem for the Character Triple Conjecture

Abstract

In this paper, we show that the Character Triple Conjecture holds for all finite groups once assumed for all quasi-simple groups. This answers the question on the existence of a self-reducing form of Dade's conjecture, a problem that was long investigated by Dade in the 1990s. Our result shows that this role is played by the Character Triple Conjecture, recently introduced by Späth, that we present here in a general form free of all previously imposed restrictions.
Paper Structure (18 sections, 42 theorems, 174 equations)

This paper contains 18 sections, 42 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. The conjecture
  3. Notation
  4. A new version of the Character Triple Conjecture
  5. Extensions of nilpotent blocks
  6. Character bijections between nilpotent Brauer correspondent blocks
  7. The Character Triple Conjecture for extensions of nilpotent blocks
  8. Equivalent forms of the Character Triple Conjecture
  9. Product chains and stable reduction for Conjecture \ref{['conj:CTC non-central Z']}
  10. Dade's correspondence over normal $p$-subgroups
  11. Proof of Theorem \ref{['thm:CTC equivalent to CTC non-central']}
  12. A cancellation theorem for the Character Triple Conjecture
  13. Brauer pairs and non-stable blocks
  14. Covering blocks of non-central normal subgroups
  15. Proof of Theorem \ref{['thm:Minimal counterexample']}
  16. ...and 3 more sections

Key Result

Theorem A

Let $G$ be a finite group and $p$ a prime number. If the Character Triple Conjecture holds at the prime $p$ for every covering group of every non-abelian simple group involved in $G$, then it holds for the group $G$ at the prime $p$.

Theorems & Definitions (87)

  • Theorem A
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Conjecture 2.3: Character Triple Conjecture
  • Lemma 2.4
  • proof
  • Conjecture 2.5: Character Triple Conjecture as in Spa17
  • Proposition 2.6
  • proof
  • ...and 77 more