The global structure theorem for finite groups with an abelian large \(p\)-subgroup (Q6142846)

Let \(G\) be a finite group and \(p \in \pi(G)\). The normalizers of non-trivial \(p\)-subgroups of \(G\) are the \(p\)-local subgroups of \(G\). The group \(G\) has characteristic \(p\) if and only if \(C_{G}(O_{p}(G)) \leq O_{p} (G)\) and \(G\) has local characteristic \(p\) if and only if all the \(p\)-local subgroups of \(G\) have characteristic \(p\). The group \(G\) is of parabolic characteristic \(p\) if and only if all the \(p\)-local subgroups containing a Sylow \(p\)-subgroup of \(G\) are of characteristic \(p\). A group \(G\) is a \(\mathcal{CK}\)-group if all composition factors are among the known simple groups. A group \(G\) is a \(\mathcal{K}_{p}\)-group, any subgroup which normalizes a non-trivial \(p\)-subgroup of \(G\) is a \(\mathcal{CK}\)-group. The paper under review is part of a program to investigate \(\mathcal{K}_{p}\)-groups of parabolic characteristic \(p\) (an important part of a revision project of the classification of the finite simple groups, see the book written by the authors [The local structure theorem for finite groups with a large \(p\)-subgroup. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1406.20017)]). Of fundamental importance for this program are the so called ``large subgroup'' (a \(p\)-subgroup \(Q\) of \(G\) is large if and only if (i) \(C_{G}(Q) \leq Q\) and (ii) \(N_{G}(U) \leq N_{G}(Q)\) for all \(1 \not = U \leq Q\)). For a prime \(p\), the local structure theorem [loc. cit.] studies finite groups \(G\) with the property that a Sylow \(p\)-subgroup \(S\) of \(G\) is contained in at least two maximal \(p\)-local subgroups. Under the additional assumptions that \(G\) is a \(\mathcal{K}_{p}\)-group and contains a large \(p\)-subgroup \(Q \leq S\), the local structure theorem partially describes the \(p\)-local subgroups of \(G\) containing \(S\), which are not contained in \(N_{G}(Q)\). In the global structure theorem, the authors describe \(N_{G}(Q)\) and, in almost all cases, the isomorphism type of the almost simple subgroup \(H\) generated by the \(p\)-local over-groups of \(S\) in \(G\) and for \(p=2\), the isomorphism type of \(G\) is determined. In the paper under review, the authors provide a reduction framework for the proof of the global structure theorem and also prove the global structure theorem when \(Q\) is abelian (Theorem C).