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Let B be a p-block of a finite group, and set m= ∑χ(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 $\CF$ 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 $\CF$-hyperfocal subgroup of P is abelian.
|Additional Information:||First published in Proceedings of the American Mathematical Society in 143 (2015), published by the American Mathematical Society|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||School of Engineering & Mathematical Sciences > Department of Mathematical Science|
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