\(G\)-algebras, Clifford theory, and the Green correspondence (Q1891493)

Let \(p\) be a prime, let \(\mathcal O\) be a suitable \(p\)-adic ring, and let \(H\) be a subgroup of a finite group \(G\). Moreover, let \(A\) be a \(G\)-algebra, let \(\alpha\), \(\beta\) be points of \(G\) and \(H\), respectively, on \(A\), and suppose that \(G_\alpha\) and \(H_\beta\) have a common defect pointed subgroup \(P_\gamma\) such that \(N_G(P_\gamma)\leq H\). The author shows that \(H_\beta \leq G_\alpha\) if and only if \(\alpha\subseteq\text{Tr}^G_H(A^H\beta A^H)\). Then he looks at this situation in more detail. Fixing \(\beta\), he studies the possible \(\alpha\), and conversely. He also indicates how his results apply to Clifford theory, the Green correspondence, the Burry-Carlson-Puig theorem and Brauer's (extended) first main theorem.