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Modular representation theory of finite groups with T.I. Sylow $ p$-subgroups


Authors: H. I. Blau and G. O. Michler
Journal: Trans. Amer. Math. Soc. 319 (1990), 417-468
MSC: Primary 20C20
DOI: https://doi.org/10.1090/S0002-9947-1990-0957081-4
MathSciNet review: 957081
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Abstract: Let $ p$ be a fixed prime, and let $ G$ be a finite group with a T.I. Sylow $ p$-subgroup $ P$. Let $ N = {N_G}(P)$ and let $ k(G)$ be the number of conjugacy classes of $ G$. If $ z(G)$ denotes the number of $ p$-blocks of defect zero, then we show in this article that $ z(G) = k(G) - k(N)$. This result confirms a conjecture of J. L. Alperin. Its proof depends on the classification of the finite simple groups. Brauer's height zero conjecture and the Alperin-McKay conjecture are also verified for finite groups with a T.I. Sylow $ p$-subgroup.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0957081-4
Keywords: Blocks of defect zero, characters of height zero, modular characters, conjugacy classes
Article copyright: © Copyright 1990 American Mathematical Society

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