Normal inclusion of Brauer pairs

For local pairs as in Brauer pair for a block, write $(Q,f)⊴(P,e)$ when $Q⊴P$, $f$ is fixed by conjugation by $P$, and ${\mathrm{Br}}_{P\mathrm{/}Q}(f)e=e$. The relative map has the fixed-algebra domain in Relative Brauer homomorphism. Define a candidate inclusion $(Q,f)\le (P,e)$ by existence of a finite chain of such normal inclusions, allowing a chain of length zero. The later order theorem proves unique descent, independence of the chain, and the partial-order properties; none is assumed here. At equal subgroups the condition is $fe=e$, which for primitive central blocks forces $f=e$.

Jacobsen, Block fusion systems and the center of the group ring, §§1.1 and 2.2, pp.3–8 and 13–18. Local argument and conventions as displayed above.