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The situation of \(Sp_ 4(4)\cdot 2\) in the sporadic simple group He. (English) Zbl 0665.20007

The main result of this paper is that Held’s simple group He contains a maximal subgroup \(Sp_ 4(4).2\). This has been known for a long time [see W. Gemmer’s Diplomarbeit (Mainz 1974)], but the present paper appears to contain the first published proof. The proof is achieved by taking two maximal subgroups \(2^ 6:3S_ 6\) of He, intersecting in a group \(2^ 6:(S_ 3\times S_ 4)\), and restricting to two subgroups \(2^ 6:(3\times A_ 5)\) intersecting in \(2^ 6:(3\times A_ 4)\). These are then used to construct the (B,N)-pair for \(Sp_ 4(4)\), which is recognized by applying a suitable characterization theorem. The final section uses Sylow’s theorem to show that the subgroup \(Sp_ 4(4)\), generated by the two groups \(2^ 6:(3\times A_ 5)\), extends to \(Sp_ 4(4).2\). For a complete determination of the maximal subgroups of He, see the paper by G. Butler [J. Algebra 69, 67-81 (1981; Zbl 0457.20024)].
Reviewer: R.A.Wilson

MSC:

20D08 Simple groups: sporadic groups
20D30 Series and lattices of subgroups

Citations:

Zbl 0457.20024
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References:

[1] B. Beisiegel , Über einfache endliche Gruppen mit Sylow 2-Gruppen der Ordnung höchstens 210 , Commun. Algebra , 5 ( 1977 ), pp. 113 - 170 . MR 424926 | Zbl 0361.20025 · Zbl 0361.20025 · doi:10.1080/00927877708822162
[2] G. Butler , The maximal subgroups of the sporadic simple group of Held , J. Alg. , 69 ( 1980 ), pp. 67 - 81 . MR 613857 | Zbl 0457.20024 · Zbl 0457.20024 · doi:10.1016/0021-8693(81)90127-7
[3] W. Gemmer , Eine Kennzeichnung der einfachen Gruppe von Held, Diplomarbeit , Universität Mainz , 1974 .
[4] E. Gerner , Ein Apartment für eine Geometrie der sporadischen einfachen Gruppe He , J. Algebra , to appear. MR 1076088 | Zbl 0712.20008 · Zbl 0712.20008 · doi:10.1016/0021-8693(90)90160-P
[5] D. Gorenstein , Finite groups , Harper and Row , New York , 1968 . MR 231903 | Zbl 0185.05701 · Zbl 0185.05701
[6] D. Held , The simple groups related to M24 , J. Alg. , 13 ( 1969 ), pp. 253 - 296 . MR 249500 | Zbl 0182.04302 · Zbl 0182.04302 · doi:10.1016/0021-8693(69)90074-X
[7] M. Suzuki , Group theory I, Springer-Verlag , Berlin , 1982 . MR 648772 | Zbl 0472.20001 · Zbl 0472.20001
[8] J. Tits , Théorème de Bruhat et sous-groupes paraboliques , C. R. Acad. Sci. Paris , 254 ( 1962 ), pp. 2910 - 2912 . MR 138706 | Zbl 0105.02201 · Zbl 0105.02201
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