A \(\{2,3\}\)-local characterization of the groups \(A_{12}\), \(\text{Sp}_8(2)\), \(F_4(2)\) and \(F_5\). (Q2498861)

There are simple groups of Lie type which have two characteristics like \(\text{PSp}(4,3)\cong\Omega^-(6,2)\). To generalize this situation to arbitrary simple groups one has to use the concept of characteristic \(p\)-type. This means that \(F^*(N)=O_p(N)\) for any \(p\)-local \(N\) of \(G\). In this language the group above is of characteristic 2-type and characteristic 3-type. The general question is which simple groups can be of characteristic 2-type and characteristic \(p\)-type for some odd prime \(p\). Groups of this type have been investigated by \textit{K. Klinger} and \textit{G. Mason} [J. Algebra 37, 362-375 (1975; Zbl 0325.20011)] as part of the classification of the finite simple groups. The paper under review is part of the GLS-project revising the classification of the finite simple groups. Hence all the groups in this paper are assumed to be \(\mathcal K\)-groups. Now the paper contributes to the \(e(G)=3\) problem. The parameter \(e(G)\) is the largest rank of an elementary Abelian \(p\)-group, \(p\) odd, which is contained in some 2-local subgroup. The authors take some prime \(p\) which is responsible for \(e(G)=3\). The question now is about characteristic 2-type and characteristic \(p\)-type at the same time. Of course the definition of characteristic \(p\)-type has been weakened in the classification project. The authors give themself two pages for explaining this. So we can just roughly state the theorem. The group is assumed to be of even type and weak \(p\)-type, i.e. just a limited set of components are allowed. These sets are dependent of the prime \(2,p\). Now the special assumptions for the theorem are, that for involutions \(x\in G\) with \(m_p(C_G(x))=3\) we have \(F^*(C_G(x))=O_2(C_G(x))\) and there is a \(2\)-local \(H\) with \(m_p(H)=3\), \(F^*(H)=O_2(H)\) and some elementary Abelian subgroup \(B\leq H\), \(|B|=p^3\) and some \(A\leq B\), \(|B:A|=p\) with \(E(C_G(A))\neq 1\). Then they prove \(p=3\) and \(G\cong A_{12}\) or \(G\) has the fusion pattern of \(F_5\) or \(G\) possesses a strongly 3-embedded subgroup \(M\cong\text{Sp}_8(2)\) or \(F_4(2)\).