A CHARACTERISATION OF NILPOTENT BLOCKS
RADHA KESSAR, MARKUS LINCKELMANN, AND GABRIEL NAVARRO

Abstract. Let B be a p-block of a finite group, and set m = ##(1)2, the sum 
taken over all height zero characters of B. Motivated by a result of M. Isaacs 
characterising p-nilpotent finite groups in terms of character degrees, we show that 
B is nilpotent if and only if the exact power of p dividing m is equal to the p-part of 
|G : P|2|P : R|, where P is a defect group of B and where R is the focal subgroup of 
P with respect to a fusion system F of B on P. The proof involves the hyperfocal 
subalgebra D of a source algebra of B. We conjecture that all ordinary irreducible 
characters of D have degree prime to p if and only if the F-hyperfocal subgroup of 
P is abelian.