Kessar, R., Linckelmann, M. & Navarro, G. (2015). A characterisation of nilpotent blocks. Proceedings of the American Mathematical Society (PROC), 143, pp. 51295138. doi: 10.1090/proc/12646
Abstract
Let B be a pblock 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 pnilpotent 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 ppart of G:P2P: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.
Publication Type:  Article 

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 
Departments:  School of Science & Technology > Mathematics 

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