On Hethelyi-Kulshammer's conjecture for principal blocks

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 2 & RADIC;p - 1. This confirms a conjecture of Hethelyi and Kulshammer (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 Maroti (2016) and Malle and Maroti (2016) on bounding the conjugacy class number and the number of p & PRIME;-degree irreducible characters of finite groups, earlier works of Broue, 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.