Covering points in permutation algebras.
Let \(K\) be a normal subgroup of a finite group \(H\), and let \(b\) be a block of \(K\) with defect group \(D\) and Brauer correspondent \(b'\) in \(N_G(D)\). \textit{M. E. Harris} and \textit{R. Knörr} [Commun. Algebra 13, 1213-1218 (1985; Zbl 0561.20002)] have shown that the Brauer homomorphism \(\text{Br}_D\) induces a defect group preserving bijection between the blocks of \(H\) covering \(b\) and the blocks of \(N_H(D)\) covering \(b'\). Now let \(\alpha\) be point of \(H\) on a \(p\)-permutation \(H\)-algebra \(A\), and let \(\beta\) be a point of \(K\) on a unitary \(H\)-subalgebra \(C\) of \(A\). The author defines what it means that \(\alpha\) covers \(\beta\). His main result then generalizes the Harris-Knörr theorem. It shows that the Brauer homomorphism \(\text{Br}_D\colon A^D\to A(D)\) induces a defect group preserving bijection between the points of \(H\) on \(A\) covering \(\beta\), with defect group in \(\mathfrak Q:=\{Q\leq H:Q\cap K=D\}\), and the points of \(N_H(D)\) on \(A(D)\) covering \(\text{Br}_D(\beta)\), with defect group in \(\mathfrak Q\).
scientific article; zbMATH DE number 6146019