Generation of fusion systems of characteristic 2-type. (Q967584)

Over the last decade a new area in group theory which is called study of fusion systems came up. Originally started by L. Puig as an area of representation theory it also has its foundation in the work of J. Alperin on the control of fusion in finite groups. But the breakthrough came with applications in some parts of topology due to Broto, Levi and Oliver. The paper under review has applications in group theory in mind. Let \(p\) be a prime and \(\mathcal F\) be a saturated fusion system on a finite \(p\)-group \(S\). The system \(\mathcal F\) is called of characteristic \(p\)-type if \(N_{\mathcal F}(U)\) is constrained for all nontrivial fully normalized subgroups \(U\) of \(S\). The author in this paper investigates such systems for \(p=2\) with an additional \(\mathcal K\)-group assumption, i.e., for \(1\neq P\leq S\) the simple composition factors of \(\Aut_{\mathcal F}(P)\) are in the list of known simple groups. As the author's aim is to use the results in a possible new proof of the classification of the finite simple groups, the assumption to be a \(\mathcal K\)-group is harmless. The paper contains one main theorem and then four corollaries. They are all very technical but here is the spirit of the main theorem. If \(\mathcal F\) is as above then with a short list of well defined exceptions (called by the author: obstructions to pushing up) the system \(\mathcal F\) is generated by \(N_{\mathcal F}(U)\), \(U\in\mathcal U\), where \(\mathcal U\) is the set of nontrivial normal subgroups \(U\) of \(S\) such that \(C_S(U)\subseteq U\) and \(O_2(\Aut_{\mathcal F}(U))=\text{Inn}(U)\). As an application he obtains the following: Let \(G\) be a finite group of characteristic 2-type, all of whose 2-local subgroups are \(\mathcal K\)-groups, then fusion in a Sylow 2-subgroup \(S\) is controlled by the normalizers in \(G\) of the subgroups \(U\) with \(C_S(U)\subseteq U\trianglelefteq S\) and \(U=O_2(N_G(U))\), again with a very short list of exceptions. In a further corollary he determines all finite groups \(G\) with \(O_2(G)=1\), all whose 2-local subgroups are \(\mathcal K\)-groups, such that a Sylow 2-subgroup is contained in a unique maximal 2-local subgroup. Another corollary is in the spirit of factorization results. Under the same assumptions as before the author proves, again with a small list of exceptions, that the fusion system \(\mathcal F\) either is generated by \(C_{\mathcal F}(\Omega_1(Z(S)))\) and \(N_{\mathcal F}(C_S(\Omega_1(Z(J(S)))))\), or for each Campbell pair \((C_1,C_2)\) for \(\mathcal F\), \(\mathcal F\) is generated by \(C_{\mathcal F}(C_1)\) and \(N_{\mathcal F}(C_2)\). A Campbell pair \((C_1,C_2)\) consists of subgroups \(1\neq C_1\leq\Omega_1(Z(S))\) and \(C_2\trianglelefteq\Aut_{\mathcal F}(\text{Baum}(S))\). The proofs use the full machinery of methods for groups of local characteristic \(p\) like Glauberman-Niles-Baumann results, TI-subgroups, pushing-up, Aschbacher blocks, weak-BN-pairs and FF-modules.