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Sum of the squares of the $p'$-character degrees

Nguyen N. Hung, J. Miquel Martínez, Gabriel Navarro

Abstract

We study the sum of the squares of the irreducible character degrees not divisible by some prime $p$, and its relationship with the the corresponding quantity in a $p$-Sylow normalizer. This leads to study a recent conjecture by E. Giannelli, which we prove for $p=2$ and in some other cases.

Sum of the squares of the $p'$-character degrees

Abstract

We study the sum of the squares of the irreducible character degrees not divisible by some prime , and its relationship with the the corresponding quantity in a -Sylow normalizer. This leads to study a recent conjecture by E. Giannelli, which we prove for and in some other cases.

Paper Structure

This paper contains 8 sections, 22 theorems, 75 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem C
  3. The McKay conjecture and inequality between character degrees
  4. Quasisimple groups
  5. Groups of Lie type in characteristic $p$
  6. Groups of Lie type in characteristic not equal to $p$
  7. Exceptional covering groups
  8. Odd-degree characters

Key Result

Lemma 2.1

Let $N\trianglelefteq\, G$ and let $\theta \in {\rm Irr}(N)$ be $G$-invariant. Then

Theorems & Definitions (46)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 36 more