Remarks on the extended Brauer quotient

On a property of the generalized Brauer pairs

“…It was generalized to the case of H-interior G-algebras, where H is normal in G and P ≤ G, in [4]. In this situation the authors restrict the constructions presented in 3.1 through 3.4 for all automorphisms ϕ ∈ Aut(P ) that satisfy ϕ(u) ∈ uH for all u ∈ P. Further generalizations of the extended Brauer quotient are given in [3]. We present them here.…”

“…Set P = P H/H. Any element ϕ ∈ Aut(P ) determines an element φ ∈ Aut( P ) with φ(u) = ϕ(u) for all u ∈ P. According to [3,Paragraph 4] as N G (P )/C H (P )-graded N G (P )-interior algebras.…”

“…The construction is useful for extending to the blocks of the normalizers of the local pointed groups the local equivalences induced by a basic Morita equivalence. The construction was then generalized in [4] to the case of H-interior G-algebras, where H is normal in G. Further remarks and properties are given in [2] and [3] for the case of G/H-graded algebras, serving for the generalization of the main rezult of [5], which was achived in [3].…”

Fusions and Clifford extensions

“…8.1. We will use the properties of the extended Brauer quotientN K B (Q) for H-interior Galgebras (see [4]) andN K A (Q) for G-graded H-interior G-algebras (see [3]), where, in both cases, K is a suitable subgroup of AutḠ(Q). Recall that, in particular,N E OG (Q) ≃ kN G (Q δ ) as E-graded algebras, with 1-component kC H (Q) as C H (Q)-interior N G (Q δ )-algebras.…”

“…Let us briefly present our setting, which is the same as in [2] and [3]. For any unexplained notions and results we refer to [17], [15] and [10].…”