Components of induced modules
The paper deals with various aspects of the Green correspondence. The interested leader should consult the paper for details. We note, however, that Theorem 4.3 is not quite correct as it stands. A revised version, suggested by the author, should read as follows: Let p be a prime and H a subgroup of the finite group G. Denote by \({\mathcal L}\) the set of p-subgroups P of H such that \(N_ G(P)\subseteq H\) and by \({\mathcal N}\) the set of p-subgroups P of H such that \(N_ G(P)\not\subseteq H\). (1) For any indecomposable H-module U with its vertices in \({\mathcal L}\), \(U^ G=V\oplus X\) where V is indecomposable with a vertex in \({\mathcal L}\) and every component of X has a vertex in \({\mathcal N}\). (2) For any indecomposable G-module V with a vertex P in \({\mathcal L}\), let \(P_ 1=P,P_ 2,...,P_ r,...,P_ n\) be representatives of each H- conjugacy class fused to P in G with \(P_ 1,...,P_ r\) the ones in \({\mathcal L}\). Then \(V_ H=W_ 1\oplus...\oplus W_ n\oplus Y\) where \(W_ i\) is indecomposable with vertex \(P_ i\) for \(i=1,...,r\), every component of \(W_ i\) has vertex \(P_ i\) for \(i=r+1,...,n\) and every component of Y has its vertices in \({\mathcal N}\). (3) For U an indecomposable H-module with vertices in \({\mathcal L}\), \(V| U^ G\) if and only if \(U| V_ H\).
scientific article; zbMATH DE number 3869572