A note on the Glauberman correspondence of \(p\)-blocks of finite \(p\)-solvable groups
Let \(p\) be a prime, and let \(S\) be a finite solvable group acting on a finite group \(G\) such that \((|G|,|S|)=1\). Let \(B\) be an \(S\)-invariant \(p\)-block of \(G\) with defect group \(D\leq C_G(S)\). \textit{A. Watanabe} has shown [J. Algebra 216, No. 2, 548-565 (1999; Zbl 0936.20006)] that all irreducible characters in \(B\) are \(S\)-invariant and that their Glauberman correspondents form a \(p\)-block \(B^*\) of \(C_G(S)\). The author proves that \(B\) and \(B^*\) are Morita equivalent whenever \(G\) is \(p\)-solvable. This fact, which generalizes a result by \textit{S. Koshitani} and \textit{G. O. Michler} [J. Algebra 243, No. 2, 504-517 (2001; Zbl 1002.20009)], was also observed by \textit{M. E. Harris} and \textit{M. Linckelmann} [Trans. Am. Math. Soc. 354, No. 9, 3435-3453 (2002; Zbl 1011.20008)].
scientific article; zbMATH DE number 1751381