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

Blau, H. I.; Michler, G. O.

doi:10.1090/S0002-9947-1990-0957081-4

Modular representation theory of finite groups with T.I. Sylow $p$-subgroups
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by H. I. Blau and G. O. Michler

Trans. Amer. Math. Soc. 319 (1990), 417-468

DOI: https://doi.org/10.1090/S0002-9947-1990-0957081-4

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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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