A 3-local identification of the alternating group of degree 8, the McLaughlin simple group and their automorphism groups. (English) Zbl 1140.20015
The authors [in Proc. Lond. Math. Soc. (3) 93, No. 2, 325-394 (2006; Zbl 1121.20010)], classified local characteristic \(p\) completions of weak \(BN\)-pairs when \(p\) is an odd prime. The outcome of this classification is that such groups are either rank 2 Lie type groups in characteristic \(p\), the weak \(BN\)-pair is of type \(\text{PSL}_3(p)\) or \(p\in\{3,5,7\}\) and the weak \(BN\)-pairs have known structure.
The groups corresponding to weak \(BN\)-pairs of type \(\text{PSL}_3(p)\) are investigated by S. J. Astill [“3-local identifications of some finite simple groups”, Univ. Birmingham, MPhill(Qual) thesis (2007)]. For the larger amalgams when \(p\in\{5,7\}\) the exceptional cases have been analyzed by the authors [J. Lond. Math. Soc., II. Ser. 69, No. 1, 128-140 (2004; Zbl 1065.20028)] and the first author and C. B. Wiedorn [Nagoya Math. J. 178, 129-149 (2005; Zbl 1085.20005)].
In the case \(p=3\) there are three different exceptional weak \(BN\)-pairs of characteristic 3. In this paper, one of these exceptional configurations is handled. The main theorem characterizes the sporadic simple group \(McL\) and \(\operatorname{Aut}(McL)\) in terms of certain 3-local data. In addition, similar characterizations of the groups \(A_8\) and \(S_8\) are obtained.
The groups corresponding to weak \(BN\)-pairs of type \(\text{PSL}_3(p)\) are investigated by S. J. Astill [“3-local identifications of some finite simple groups”, Univ. Birmingham, MPhill(Qual) thesis (2007)]. For the larger amalgams when \(p\in\{5,7\}\) the exceptional cases have been analyzed by the authors [J. Lond. Math. Soc., II. Ser. 69, No. 1, 128-140 (2004; Zbl 1065.20028)] and the first author and C. B. Wiedorn [Nagoya Math. J. 178, 129-149 (2005; Zbl 1085.20005)].
In the case \(p=3\) there are three different exceptional weak \(BN\)-pairs of characteristic 3. In this paper, one of these exceptional configurations is handled. The main theorem characterizes the sporadic simple group \(McL\) and \(\operatorname{Aut}(McL)\) in terms of certain 3-local data. In addition, similar characterizations of the groups \(A_8\) and \(S_8\) are obtained.
Reviewer: Anatoli Kondrat’ev (Ekaterinburg)
MSC:
| 20D08 | Simple groups: sporadic groups |
| 20D05 | Finite simple groups and their classification |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |
| 20E42 | Groups with a \(BN\)-pair; buildings |
| 51E24 | Buildings and the geometry of diagrams |
| 20D45 | Automorphisms of abstract finite groups |
Keywords:
finite groups; McLaughlin simple group; 3-local characterizations; local characteristic \(p\) completions; weak \(BN\)-pairsSoftware:
MagmaReferences:
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