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Minimal heights and defect groups with two character degrees

Gunter Malle, Alexander Moretó, Noelia Rizo

Abstract

Conjecture A of \cite{EM14} predicts the equality between the smallest positive height of the irreducible characters in a $p$-block of a finite group and the smallest positive height of the irreducible characters in its defect group. Hence, it can be seen as a generalization of Brauer's famous height zero conjecture. One inequality was shown to be a consequence of Dade's Projective Conjecture. We prove the other, less well understood, inequality for principal blocks when the defect group has two character degrees.

Minimal heights and defect groups with two character degrees

Abstract

Conjecture A of \cite{EM14} predicts the equality between the smallest positive height of the irreducible characters in a -block of a finite group and the smallest positive height of the irreducible characters in its defect group. Hence, it can be seen as a generalization of Brauer's famous height zero conjecture. One inequality was shown to be a consequence of Dade's Projective Conjecture. We prove the other, less well understood, inequality for principal blocks when the defect group has two character degrees.
Paper Structure (6 sections, 24 theorems, 18 equations)

This paper contains 6 sections, 24 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Linear and permutation groups
  3. Results on almost simple groups
  4. Miscellaneous results
  5. Almost simple groups of Lie type
  6. Proof of Theorem 1

Key Result

Theorem 1

Let $p$ be a prime and let $G$ be a finite group. If a Sylow $p$-subgroup $P$ of $G$ has two character degrees, then ${{\operatorname{mh}}}(B_0(G))\leq {{\operatorname{mh}}}(P)$.

Theorems & Definitions (46)

  • Theorem 1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 36 more