Domination of blocks, fusion systems and hyperfocal subgroups (Q2292838)

Let \(k\) be an algebraically closed field of characteristic \(p>0\), let \(G\) be a finite group, and let \(b\) be a block (idempotent) of the group algebra \(kG\). Moreover, let \((D,e)\) be a maximal \(b\)-Brauer pair, and let \(\mathcal{F}\) denote the corresponding fusion system of \(b\) on the defect group \(D\) of \(b\). The hyperfocal subgroup of \(\mathcal{F}\) is defined by \[ \mathrm{hyp}(\mathcal{F}) := \langle u \phi(u^{-1}):u \in Q \le D, \; \phi \in \mathrm{O}^p(\mathrm{Aut}_{\mathcal F}(Q))\rangle. \] The block \(b\) is called inertial if there exists a basic Morita equivalence between \(b\) and its Brauer correspondent in \(kN_G(D)\). It is called nilpotent if \(\mathrm{hyp}(\mathcal{F}) = 1\). Now let \(P\) be a normal \(p\)-subgroup of \(G\), and let \(\pi: kG \longrightarrow k{\bar G}\) be the canonical epimorphism where \({\bar G} := G/P\). One of the main results of the paper shows: \begin{itemize} \item[(i)] If \(b\) is nilpotent then \(\pi(b)\) is a nilpotent block of \(k{\bar G}\); \item[(ii)] If \(D = Q \times P\) with \(\mathrm{hyp}(\mathcal{F}) \le Q\) then \(b\) is inertial if and only if \(\pi(b)\) is an inertial block of \(k{\bar G}\). \end{itemize} The authors also investigate connections between fusion systems and hyperfocal subgroups of the block \(b\) of \(kG\) and of the block \(\pi(b)\) of \(k{\bar G}\) in the situation where \(G/C_G(P)\) is a \(p\)-group.