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On Héthelyi–Külshammer's conjecture for principal blocks

Nguyen Ngoc Hung and A. A. Schaeffer Fry

Vol. 17 (2023), No. 6, 1127–1151
Abstract

We prove that the number of irreducible ordinary characters in the principal p-block of a finite group G of order divisible by p is always at least 2p 1. This confirms a conjecture of Héthelyi and Külshammer (2000) for principal blocks and provides an affirmative answer to Brauer’s problem 21 (1963) for principal blocks of bounded defect. Our proof relies on recent works of Maróti (2016) and Malle and Maróti (2016) on bounding the conjugacy class number and the number of p-degree irreducible characters of finite groups, earlier works of Broué, Malle and Michel (1993) and Cabanes and Enguehard (2004) on the distribution of characters into unipotent blocks and e-Harish-Chandra series of finite reductive groups, and known cases of the Alperin–McKay conjecture.

Keywords
finite groups, principal blocks, characters, Héthelyi–Külshammer conjecture, Alperin–McKay conjecture
Mathematical Subject Classification
Primary: 20C15, 20C20, 20C33, 20D06
Milestones
Received: 22 April 2021
Revised: 16 January 2022
Accepted: 6 July 2022
Published: 26 May 2023
Authors
Nguyen Ngoc Hung
Department of Mathematics
Buchtel College of Arts and Sciences
The University of Akron
Akron, OH
United States
A. A. Schaeffer Fry
Department of Mathematics and Statistics
Metropolitan State University of Denver
Denver, CO
United States

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