On H\'ethelyi-K\"ulshammer's conjecture for principal blocks
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 H\'ethelyi and K\"ulshammer for principal blocks and provides an affirmative answer to Brauer's Problem 21 for principal blocks of bounded defect. Our proof relies on recent works of Mar\'oti and Malle-Mar\'oti on bounding the conjugacy class number and the number of p'-degree irreducible characters of finite groups, earlier works of Brou\'e-Malle-Michel and Cabanes-Enguehard 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.
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