Abstract
We prove that a periodic group is locally finite, given that each of its finite subgroups lies in a subgroup isomorphic to a finite simple group G2 of Lietypeovera field of odd characteristic.
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References
R. W. Carter, Simple Groups of Lie Type (John Wiley & Sons, London, 1972).
V. V. Belyaev, “Locally Finite Chevalley Groups,” in Investigations in Group Theory (USC AS USSSR, Sverdlovsk, 1984), pp. 39–50 [in Russian].
A. V. Borovik, “Embeddings of finite Chevalley groups and periodic linear groups,” Sib. Math. J. 24 (6), 26–35 (1983) [Sib. Math. J. 24 (6), 843–851 (1983)].
B. Hartley and G. Shute, “Monomorphisms and direct limits of finite groups of Lie type,” Quart. J. Math. Oxford Ser. (2) 35 (137), 49–71 (1984).
S. Thomas, “The classification of the simple periodic linear groups,” Arch. Math. (Basel) 41 (2), 103–116 (1983).
M. J. Larsen and R. Pink, “Finite subgroups of algebraic groups,” J. Amer. Math. Soc. 24 (4), 1105–1158 (2011).
A. K. Shlepkin, “On some periodic groups saturated by finite simple groups,” Mat. Tr. 1 (1), 129–138 (1998) [Sib. Adv. in Math. 9 (2), 100–108 (1999)].
A. G. Rubashkin and K. A. Filippov, “Periodic groups saturated with the groups L 2(p n),” Sib. Math. J. 46 (6), 1388–1392 (2005) [Sib. Math. J. 46 (6), 1119–1122 (2005)].
D. V. Lytkina and A. A. Shlepkin, “Periodic groups saturated with finite simple groups of types U 3 and L 3,” Algebra and Logic 55 (4), 441–448 (2016) [Algebra and Logic 55 (4), 289–294 (2016)].
K. A. Filippov, Groups Saturated with Finite Non-Abelian Simple Groups and Their Extensions, Cand. Sci. (Phys.–Math.) Dissertation (Krasnoyarsk, 2005) [in Russian].
K. A. Filippov, “On periodic groups saturated by finite simple groups,” Sib. Math. J. 53(2), 430–438 (2012) [Sib. Math. J. 53 (2), 345–351 (2012)].
D. Gorenstein, Finite Groups (Chelsea Publ., New York, 1980).
J. H. Conway, R.T. Curtis, S. P. Norton, R.A. Parker and R. A. Wilson, Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Group (Clarendon Press, Oxford, 1985).
D. Gorenstein and K. Harada, “Finite simple groups of low 2-rank and the families G 2(q), D 24 , q odd,” Bull. Amer. Math. Soc. 77 (6), 829–862 (1971).
J. N. Bray, D. F. Holt, and C. M. Roney-Dougal, The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, in London Math. Soc. Lecture Note Ser. (Cambridge Univ. Press, Cambridge, 2013), Vol. 407.
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Zhu, X., Lytkina, D.V. & Mazurov, V.D. Characterization of Locally Finite Simple Groups of Type G2 over Fields of Odd Characteristics in the Class of Periodic Groups. Math Notes 105, 513–518 (2019). https://doi.org/10.1134/S0001434619030234
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DOI: https://doi.org/10.1134/S0001434619030234