It is proved that a periodic group whose element orders do not exceed 6 either is a locally finite or is group of exponent 5.
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References
R. Brandl and W. J. Shi, “A characterization of finite simple groups with Abelian Sylow 2-subgroups,” Ric. Mat., 42, No. 1, 193–198 (1993).
Unsolved Problems in Group Theory, The Kourovka Notebook, 18th edn., Institute of Mathematics SO RAN, Novosibirsk (2014), http://www.math.nsc.ru/∼alglog/18kt.pdf.
W. Burnside, “On an unsettled question in the theory of discontinuous groups,” Q. J. Pure Appl. Math., 33, 230–238 (1902).
B. H. Neumann, “Groups whose elements have bounded orders,” J. London Math. Soc., 12, 195–198 (1937).
I. N. Sanov, “Solution of the Burnside problem for period 4,” Uch. Zap. LGU, Ser. Mat., 10, 166–170 (1940).
M. Hall, Jr., “Solution of the Burnside problem for exponent six,” Ill. J. Math., 2, 764–786 (1958).
M. F. Newman, “Groups of exponent dividing seventy,” Math. Sci., 4, 149–157 (1979).
N. D. Gupta and V. D. Mazurov, “On groups with small orders of elements,” Bull. Aust. Math. Soc., 60b, No. 5, 197–205 (1999).
V. D. Mazurov, “Groups of exponent 60 with prescribed orders of elements,” Algebra and Logic, 39, No. 3, 189–198 (2000).
E. Jabara, “Fixed point free actions of groups of exponent 5,” J. Aust. Math. Soc., 77, No. 3, 297–304 (2004).
V. D. Mazurov and A. S. Mamontov, “On periodic groups with small orders of elements,” Sib. Math. J., 50, No. 2, 316–321 (2009).
A. S. Mamontov, “Groups of exponent 12 without elements of order 12,” Sib. Math. J., 54, No. 1, 114–118 (2013).
The GAP Group, GAP—Groups, Algorithms, Programming—a System for Computational Discrete Algebra, vers. 4.7.5 (2014); http://www.gap-system.org.
P. Hall and G. Higman, “The p-length of p-soluble groups and reduction theorems for Burnside’s problem,” Proc. London Math. Soc., III. Ser., 6, No. 1, 1–42 (1956).
V. P. Shunkov, “On periodic groups with an almost regular involution,” Algebra and Logic, 11, No. 4, 260–272 (1972).
M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], 4th edn., Nauka. Fizmatlit, Moscow (1996).
D. V. Lytkina, L. R. Tukhvatullina, and K. A. Filippov, “The periodic groups saturated by finitely many finite simple groups,” Sib. Math. J., 49, No. 2, 317–321 (2008).
V. D. Mazurov, “Recognition of finite simple groups S4(q) by their element orders,” Algebra and Logic, 41, No. 2, 93–110 (2002).
V. P. Shunkov, “On a class of p-groups,” Algebra and Logic, 9, No. 4, 291–297 (1970).
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(D. V. Lytkina) Supported by RFBR, grant Nos. 13-01-00505 and 14-01-90013.
(V. D. Mazurov, A. S. Mamontov and E. Jabara) The work is supported by Russian Science Foundation (project 14-21-00065).
Translated from Algebra i Logika, Vol. 53, No. 5, pp. 570–586, September-October, 2014.
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Lytkina, D.V., Mazurov, V.D., Mamontov, A.S. et al. Groups Whose Element Orders do not Exceed 6. Algebra Logic 53, 365–376 (2014). https://doi.org/10.1007/s10469-014-9297-2
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DOI: https://doi.org/10.1007/s10469-014-9297-2