AbstractLet k be an algebraically closed field of prime characteristic p.
Let G be a finite group, let N be a normal subgroup of G, and let c be a G-stable block of kN so that {(kN)c} is a p-permutation G-algebra.
As in Section 8.6 of [M. Linckelmann,
The Block Theory of finite Group Algebras: Volume 2,
London Math. Soc. Stud. Texts 92,
Cambridge University, Cambridge, 2018],
a {(G,N,c)}-Brauer pair {(R,f_{R})} consists of a p-subgroup R of G and a block {f_{R}} of {(kC_{N}(R))}.
If Q is a defect group of c and {f_{Q}\in\operatorname{\textit{B}\ell}(kC_{N}(Q))}, then {(Q,f_{Q})} is a {(G,N,c)}-Brauer pair.
The {(G,N,c)}-Brauer pairs form a (finite) poset.
Set {H=N_{G}(Q,f_{Q})} so that {(Q,f_{Q})} is an {(H,C_{N}(Q),f_{Q})}-Brauer pair.
We extend Lemma 8.6.4 of the above book to show that if {(U,f_{U})} is a maximal {(G,N,c)}-Brauer pair containing {(Q,f_{Q})}, then {(U,f_{U})} is a maximal {(H,C_{N}(c),f_{Q})}-Brauer pair containing {(Q,f_{Q})} and conversely.
Our main result shows that the subcategories of {\mathcal{F}_{(U,f_{U})}(G,N,c)} and {\mathcal{F}_{(U,f_{U})}(H,C_{N}(Q),f_{Q})} of objects between and including {(Q,f_{Q})} and {(U,f_{U})} are isomorphic.
We close with an application to the Clifford theory of blocks.