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Let $B$ be a $p$-block of a finite group, and set $m=$ $\sum \chi(1)^2$, 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.
[1] M. Aschbacher, R. Kessar and B. Oliver, Fusion systems in algebra and topology, London Mathematical Society Lecture Note Series, Vol. 391, Cambridge University Press, Cambridge, 2011.
[2] M. Brou´e and L. Puig, Characters and Local Structure in G-Algebras, J. Algebra 63 (1980), 306-317.
[3] D. A. Craven, The Theory of Fusion Systems, Cambridge Studies in Advanced Mathematics, Vol. 131, Cambridge University Press, Cambridge, 2011.
[4] A. Diaz, A. Glesser, S. Park, and R. Stancu, Tate's and Yoshida's theorems on control of transfer for fusion systems. J. Lond. Math. Soc. (2) 84 (2011), no. 2, 475-494.
[5] Y. Fan, Hyperfocal subalgebras of blocks and computation of characters. J. Algebra 322 (2009), no. 10, 3681-3692.
[6] S. M. Gagola Jr., I. M. Isaacs, Transfer and Tate's theorem, Arch. Math. (Basel) 91 (2008) 300306.
[7] D. Gluck and T. R. Wolf, Brauer's height zero conjecture for p-solvable groups. Trans. Amer. Math. Soc. 282 (1984), 137-152.
[8] M. E. Harris and M. Linckelmann, On the Glauberman and Watanabe correspondences for blocks of finite p-solvable groups, Trans. Amer. Math. Soc. 354 (2002), 3435-3453.
[9] I. M. Isaacs, Recovering information about a group from its complex group algebra, Arch. Math. 46 (1986), 293-295.
[16] L. Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), 77-116.
[17] L. Puig The hyperfocal subalgebra of a block, Invent. Math. 141 (2000), 365-397.
[18] G. R. Robinson, On the focal defect group of a block, characters of height zero, and lower defect group multiplicities. J. Algebra 320 (2008), no. 6, 2624-2628.
[19] J. Th´evenaz, Duality in G-algebras, Math. Z. 200 (1988) 47-85. School of Engineering and Mathematical Sciences, City University London EC1V 0HB, Great Britain E-mail address: School of Engineering and Mathematical Sciences, City University London EC1V 0HB, Great Britain E-mail address: Departament d'A` lgebra, Universitat de Val`encia, Dr. Moliner 50, 46100 Burjassot, Spain. E-mail address: