On the Glauberman and Watanabe correspondences for blocks of finite 𝑝-solvable groups

Harris, M.; Linckelmann, M.

doi:10.1090/S0002-9947-02-02990-2

On the Glauberman and Watanabe correspondences for blocks of finite $p$-solvable groups
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by M. E. Harris and M. Linckelmann PDF

Trans. Amer. Math. Soc. 354 (2002), 3435-3453 Request permission

Abstract:

If $G$ is a finite $p$-solvable group for some prime $p$, $A$ a solvable subgroup of the automorphism group of $G$ of order prime to $\vert G\vert$ such that $A$ stabilises a $p$-block $b$ of $G$ and acts trivially on a defect group $P$ of $b$, then there is a Morita equivalence between the block $b$ and its Watanabe correspondent $w(b)$ of $C_{G}(A)$, given by a bimodule $M$ with vertex $\Delta P$ and an endo-permutation module as source, which on the character level induces the Glauberman correspondence (and which is an isotypy by Watanabe’s results).

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