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Periodic groups with prescribed element orders. (English) Zbl 1459.20033

Summary: Some recent results obtained with the participation of the author are discussed.
1. Known results on the structure of periodic groups with given spectra.
2. On groups of period 24.
3. The local finiteness of some groups with elementary abelian centralizers of involutions.
4. Recognition of finite simple groups by orders and spectra.

MSC:

20F50 Periodic groups; locally finite groups
20E25 Local properties of groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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[1] [1] F. Levi, B.L. van der Waerden, “Über eine besondere Klasse von Gruppen”, Abh. Math. Semin. Hamburg Univ., 9 (1932), 154–158. · Zbl 0005.38507 · doi:10.1007/BF02940639
[2] [2] F.V. Levi, “Groups in which the commutator operations satisfy certain algebraical conditions”, J. Indian Math. Soc., 6 (1942), 87–97 · Zbl 0061.02606
[3] [3] I.N. Sanov, “Solution of the Burnside problem for exponent 4 (in Russian)”, Uchenye zap. Leningr. gos. universiteta, matem. seriya, 10 (1940), 166–170
[4] [4] Yu.P. Razmyslov, “The Hall-Higman problem”, Izv. AN SSSR. Ser. matem., 42:4 (1978), 833–847 (in Russian) · Zbl 0394.20030
[5] [5] M. Hall Jr., “Solution of the Burnside problem for exponent six”, Illinois J. Math., 2 (1958), 764–786 · Zbl 0083.24801
[6] [6] S.I. Adyan, The Burnside problem and identities in groups, Nauka, Moscow, 1975 (in Russian) · Zbl 0306.20045
[7] [7] A.Yu. Ol’shanskii, The geometry of defining relations in groups, Nauka, Moscow, 1989 (in Russian)
[8] [8] S.V. Ivanov, “The free Burnside groups of sufficiently large exponents”, Internat. J. Algebra Comput., 4 (1994), 3–308 · Zbl 0822.20044 · doi:10.1142/S0218196794000026
[9] [9] I.G. Lysenok, “Infinite Burnside groups of even exponents”, Izv. RAN. Ser. matem., 60:3 (1996), 3–224 (in Russian) · Zbl 0926.20023
[10] [10] B.H. Neumann, “Groups whose elements have bounded orders”, J. London Math. Soc., 12 (1937), 195–198 · Zbl 0016.39303 · doi:10.1112/jlms/s1-12.2.195
[11] [11] D.D. Lytkina, “The structure of a group whose element orders are at most 4”, Sib. matem. zhurn., 48:2 (2007), 353–358 (in Russian) · Zbl 1154.20036
[12] [12] E. Jabara., “Fixed point free action of groups of exponent 5”, J. Austral. Math. Soc., 77 (2004), 297–304 · Zbl 1106.20031 · doi:10.1017/S1446788700014440
[13] [13] M.F. Newman, “Groups of exponent dividing seventy”, Math. Scientist., 4 (1979), 149–157 · Zbl 0654.20038
[14] [14] A.Kh. Zhurtov, V.D. Mazurov, “On recognizing finite simple groups \(L_2(2^m)\) in the class of all groups”, Sib. matem. zhurn., 40:1 (1999), 75–78 (in Russian) · Zbl 0941.20010
[15] [15] V.D. Mazurov, “On infinite groups with abelian centralizers of involutions”, Algebra i logika, 39:1 (2000), 74–86 (in Russian) · Zbl 1093.68611
[16] [16] A.I. Sozutov, “On the structure of the noninvariant multiplier in certain Frobenius groups”, Sib. matem. zhurn., 35:4 (1994), 893–901 (in Russian) · Zbl 0851.20039
[17] [17] N.D. Gupta, V.D. Mazurov, “On groups with small orders of elements”, Bull. Austral. Math. Soc., 60 (1999), 197–205 · Zbl 0939.20043 · doi:10.1017/S0004972700036339
[18] [18] V.D. Mazurov, “On groups of exponent 60 with prescribed orders of elements”, Algebra i logika, 39:3 (2000), 329–346 (in Russian) · Zbl 0979.20037
[19] [19] A.Kh. Zhurtov, “On regular automorphisms of order 3 and Frobenius pairs”, Sib. matem. zhurn., 41:2 (2000), 329–338 (in Russian) · Zbl 0956.20036 · doi:10.1007/BF02674603
[20] [20] A.Kh. Zhurtov, “On quadratic automorphisms of abelian groups”, Algebra i logika, 39:3 (2000), 320–328 (in Russian) · Zbl 0980.20049
[21] [21] V.D. Mazurov, A.S. Mamontov, “On periodical groups with elements of small orders”, Sib. matem. zhurn., 50:2 (2009), 397–404 (in Russian)
[22] [22] D.V. Lytkina, A.A. Kuznetsov, “Recognizability by spectrum of the group \(L_2(7)\) in the class of all groups”, Siberian Electronic Mathematical Reports, 4 (2007), 136–140 · Zbl 1134.20009
[23] [23] D. Sonkin, On groups of large exponents \(n\) and \(n\)-periodic products, PhD Thesis, Vandebilt Univ., Nashville, Tennessee, 2005
[24] [24] Shi W., “A new characterization of the sporadic simple groups”, Group Theory, Proc. 1987 Singapore Group Theory Conference, Walter de Gruyter, Berlin–New York, 1989, 531–540
[25] [25] Bi J. and Shi W., “A characteristic property for each finite projective special linear group”, Lecture Notes in Math., 1456, Springer–Verlag, Berlin, 1990, 171–180 · Zbl 0718.20009
[26] [26] Bi J. and Shi W., “A characterization of Suzuki-Ree groups”, Science in China (Ser. A), 34:1 (1991), 14–19 · Zbl 0736.20011
[27] [27] Bi J. and Shi W., “A new characterization of the alternating groups”, Southeast Asian Bull. Math., 16:1 (1992), 81–90 · Zbl 0790.20030
[28] [28] Shi W., “The pure quantitative characterization of finite simple groups (I)”, Progress in Natural Science, 4:3 (1994), 316–326
[29] [29] Cao H. and Shi W., “The pure quantitative characterization of finite projective special unitary groups”, Science in China (Ser. A), 45:6 (2002), 761–772 · Zbl 1099.20507
[30] [30] Shi W. and Xu M., “Pure quantitative characterization of finite simple groups \(^2D_n(q)\) and \(D_l(q)\) (\(l\) odd)”, Algebra Colloq., 10:3 (2003), 427–443 · Zbl 1034.20018
[31] [31] Williams J.S., “Prime graph components of finite groups”, J. Algebra, 69:2 (1981), 487–513 · Zbl 0471.20013 · doi:10.1016/0021-8693(81)90218-0
[32] [32] Kondrat’ev A.S., “On prime graph components of finite simple groups”, Math. USSR-Sb., 67:1 (1990), 235–247 · Zbl 0698.20009 · doi:10.1070/SM1990v067n01ABEH001363
[33] [33] Vasil’ev A.V., “On connection between the structure of a finite group and properties of its prime graph”, Siberian Math. J., 46:3 (2005), 396–404 · doi:10.1007/s11202-005-0042-x
[34] [34] Vasiliev A.V. and Vdovin E.P., “An adjacency criterion for the prime graph of a finite simple group”, Algebra and Logic, 44:6 (2005), 381–406 · Zbl 1104.20018 · doi:10.1007/s10469-005-0037-5
[35] [35] Mazurov V.D., “Recognition of finite simple groups \(S_4(q)\) by their element orders”, Algebra and Logic, 41:2 (2002), 93–110 · Zbl 1067.20016 · doi:10.1023/A:1015356614025
[36] [36] Vasil’ev A.V. and Grechkoseeva M.A., “On recognition by spectrum of finite simple linear groups over fields of characteristic 2”, Siberian Math. J., 46:4 (2005), 593–600 · doi:10.1007/s11202-005-0060-8
[37] [37] Vasilyev A.V. and Grechkoseeva M.A., “Recognition by spectrum for finite simple linear groups of small dimensions over fields of characteristic 2”, Algebra and Logic, 47:5 (2008), 314–320 · Zbl 1155.20025 · doi:10.1007/s10469-008-9026-9
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