Broué's abelian defect group conjecture for blocks with cyclic hyperfocal subgroups (Q6614913)

Assume that \(b\) is a \(p\)-block of the group algebra \(\mathcal OG\) of a finite group \(G\) with defect pointed group \(P_\gamma\), that is, \(P\) is a defect group of \(b\) and \(\gamma\) is a \(P\)-source point of \(b\), namely, for \(i\in\gamma\), the interior \(P\)-algebra \(i\mathcal OGi\) is a (so-called) \(P\)-source algebra of \(b\). The notion, which generalizes Richard Brauer's classical theory, is due to \textit{L. Puig} [Math. Z. 176, 265--292 (1981; Zbl 0464.20007)], who defines also a hyperfocal subgroup \(D:=\mathfrak{hyp}_G(P_\gamma)\) of \(P_\gamma\), namely \(D:=\langle[L,Q]\midQ_\delta\in{\mathcal{LPS}}(P_\gamma), L\in\mathcal S_{p'}(N_G(Q_\delta))\rangle\), where \(\mathcal{LPS}(P_\gamma)\) is the set of all local pointed subgroups \(Q_\delta\leq P_\gamma\) on \(\mathcal OG\) and \(\mathcal S_{p'}(N_G(Q_\delta))\) is the set of all \(p'\)-subgroups of \(N_G(Q_\delta):=\{g\in N_G(Q)\mid g\delta g^{-1}=\delta \}\).\N\N\NOne of the main results of the paper under review says the following: Assume that the defect group \(P\) is abelian and the hyperfocal subgroup \(D\) of \(P_\gamma\) is cyclic. Then, all the three block algebras \(\mathcal OGb\), \(\mathcal ON_G(P)b_0\) and \(\mathcal ON_G(D)c\) are Rickard equivalent, where \(b_0\) and \(c\) are the Brauer corresponding block idempotents (blocks) of \(b\) in \(N_G(P)\) and \(N_G(D)\), respectively. As a byproduct, the authors also prove that Broué's abelian defect group conjecture holds, namely the block algebras \(\mathcal OGb\) and \(\mathcal ON_G(P)b_0\) are derived equivalent, provided \(P\) is abelian and the hyperfocal subgroup \(D\) above is cyclic. Not only \textit{L. Puig}'s results [On the local structure of Morita and Rickard equivalences between Brauer blocks. Basel: Birkhäuser (1999; Zbl 0929.20012); Invent. Math. 141, No. 2, 365--397 (2000; Zbl 0957.20007)] but also \textit{A. Watanabe}'s results\N[RIMS Kokyuroku 1140, 76--79 (2000; Zbl 0968.20502); J. Algebra 416, 167--183 (2014; Zbl 1328.20018)] play an important role.