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7.6 (1980)

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Describe the infinite simple locally finite groups with a Chernikov Sylow 2-subgroup. In particular, are such groups the Chevalley groups over locally finite fields of odd characteristic?

Contributor: V. V. Belyaev, N. F. Sesekin

Progress

No, as every countably infinite locally finite $p$-group can be embedded as a maximal $p$-subgroup of the simple Hall’s universal group $U$, the unique countable existentially closed group in the class of all locally finite groups (K. Hickin, Proc. London Math. Soc. (3), 52, no. 1 (1986), 53–72). But if all Sylow 2-subgroups are Chernikov, then this is true mod CFSG (O. H. Kegel, Math. Z., 95 (1967), 169–195; V. V. Belyaev, in: Investigations in Group Theory, Sverdlovsk, UNC AN SSSR, 1984, 39–50 (Russian); A. V. Borovik, Siberian Math. J., 24, no. 6 (1983), 843–851; B. Hartley, G. Shute, Quart. J. Math. Oxford (2), 35 (1984), 49–71; S. Thomas, Arch. Math., 41 (1983), 103–116). When all Sylow 2-subgroups are Chernikov, the same result, without using CFSG, follows from (M. J. Larsen, R. Pink, J. Amer. Math. Soc., 24 (2011), 1105–1158 (also known as a preprint of 1998)).

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