## Groups with given element orders.(English)Zbl 07325161

Summary: This paper is a survey of some results and open problems about the structure of (mostly infinite) periodic groups with a given set of element orders. It is based on a talk of authors given on the conference “Algebra and Logic: Theory and Application” dedicated to the 80-th anniversary of V. P. Shunkov (Krasnoyarsk, July 21–27, 2013).

### MSC:

 20-XX Group theory and generalizations 00-XX General and overarching topics; collections

### Keywords:

spectrum; exponent; periodic group; locally finite group
Full Text:

### References:

 [1] C. Adelmann, E. H.-A. Gerbracht, “Letters from William Burnside to Robert Fricke: automorphic functions, and the emergence of the Burnside Problem”, Arch. Hist. Exact Sci., 63:1 (2009), 33-50 · Zbl 1172.01008 [2] M. Hall, “Solution of the Burnside problem for exponent six”, Illinois J. Math., 2 (1958), 764-786 · Zbl 0083.24801 [3] W. Burnside, “On an unsettled question in the theory of discontinuous groups”, Quart. J. Pure Appl. Math., 33 (1902), 230-238 · JFM 33.0149.01 [4] S. I. Adyan, “The Burnside problem and related questions”, Russian Math. Surveys, 65:5 (2010), 805-855 · Zbl 1230.20001 [5] M. A. Hilton, An introduction to the theory of groups of finite order, Clarendon Press, Oxford, 1908 · JFM 39.0193.01 [6] W. Burnside, “On groups in which every two conjugate operations are permutable”, Proc. London Math. Soc., 35 (1903), 28-37 · JFM 34.0154.01 [7] C. Hopkins, “Finite groups in which conjugate operations are commutative”, Amer. J. Math., 51 (1929), 35-41 · JFM 55.0081.10 [8] F. Levi, “Groups in which the commutator operations satisfy certain algebraic conditions”, J. Indian Math. Soc., 6 (1942), 166-170 [9] F. Levi, B. van der Waerden, “Über eine besondere Klasse von Gruppen”, Abh. Math. Semin., Hamburg Univ., 9 (1932), 157-158 · JFM 58.0125.02 [10] I. N. Sanov, “Solution of Burnside”s problem for exponent 4”, Leningrad State University Annals (Uchenye Zapiski) Math. Ser., 1940, no. 10, 166-170 (in Russian) · Zbl 0061.02506 [11] Yu. P. Razmyslov, “The Hall-Higman problem”, Math. USSR - Izv., 13:1 (1979), 133-146 · Zbl 0422.20028 [12] A. I. Kostrikin, Around Burnside, Ergeb. Math. Grenzgeb. (3), 20, Springer-Verlag, New York-Berlin-Heidelberg, 1990 · Zbl 0702.17001 [13] P. Hall, G. Higman, “On the $$p$$-length of $$p$$-soluble groups and reduction theorems for Burnside”s problem”, Proc. London Math. Soc., 6:3 (1956), 1-42 · Zbl 0073.25503 [14] M. F. Newman, “Groups of exponent six”, Computational group theory (Durham, 1982), Academic Press, London, 1984, 39-41 [15] I. G. Lysenok, “Proof of a theorem of M. Hall concerning the finiteness of the groups $$B(m,6)$$”, Math. Notes, 41:3 (1987), 241-244 · Zbl 0626.20028 [16] P. S. Novikov, “On periodic groups”, Dokl. Akad. Nauk, 127 (1959), 749-752 (in Russian) · Zbl 0119.02202 [17] P. S. Novikov, S. I. Adian, “On infinite periodic groups. I”, Izv. Akad. Nauk SSSR, Ser. mat., 32:1 (1968), 212-244 (in Russian) · Zbl 0194.03301 [18] P. S. Novikov, S. I. Adian, “On infinite periodic groups. II”, Izv. Akad. Nauk SSSR, Ser. mat., 32:2 (1968), 251-524 (in Russian) · Zbl 0194.03301 [19] P. S. Novikov, S. I. Adian, “On infinite periodic groups. III”, Izv. Akad. Nauk SSSR, Ser. mat., 32:3 (1968), 709-731 (in Russian) · Zbl 0194.03301 [20] S. I. Adian, The Burnside problem and identities in groups, Springer-Verlag, Berlin-New York, 1978 [21] S. V. Ivanov, “The free Burnside groups of sufficiently large exponents”, Internat. J. Algebra Comput., 4 (1994), 3-308 · Zbl 0822.20044 [22] I. G. Lysenok, “Infinite Burnside groups of even exponent”, Izv. Math., 60:3 (1996), 453-654 · Zbl 0926.20023 [23] B. H. Neumann, “Groups whose elements have bounded orders”, J. London Math. Soc., 12 (1937), 195-198 · Zbl 0016.39303 [24] D. V. Lytkina, “Structure of a group with elements of order at most 4”, Siberian Math. J., 48:2 (2007), 283-287 · Zbl 1154.20036 [25] B. H. Neumann, “Groups with automorphisms that leave only the neutral element fixed”, Arch. Math., 7 (1956), 1-5 · Zbl 0070.02203 [26] A. Kh. Zhurtov, “Regular automorphisms of order 3 and Frobenius pairs”, Siberian Math. J., 41:2 (2000), 268-275 · Zbl 0956.20036 [27] E. Jabara, “Fixed point free action of groups of exponent 5”, J. Austral. Math. Soc., 77 (2004), 297-304 · Zbl 1106.20031 [28] E. Jabara, “Free actions of groups of exponent 5”, Algebra and Logic, 50:5 (2011), 466-469 · Zbl 1257.20040 [29] A. Kh. Zhurtov, D. V. Lytkina, V. D. Mazurov, A. I. Sozutov, “Periodic groups acting freely on Abelian groups”, Tr. IMM UrB RAS, 19, no. 3, 2013, 136-143 (in Russian) · Zbl 1307.20027 [30] E. Jabara, P. Mayr, “Frobenius complements of exponent dividing $$2^m\cdot9$$”, Forum Mathematicum, 21:1 (2009), 217-220 · Zbl 1177.20041 [31] D. V. Lytkina, “Periodic groups acting freely on abelian groups”, Algebra and Logic, 49:3 (2010), 256-264 · Zbl 1216.20026 [32] V. P. Shunkov, “On a class of $$p$$-groups”, Algebra i logika, 9:4 (1970), 484-496 (in Russian) · Zbl 0237.20032 [33] V. M. Busarkin, Yu. M. Gorchakov, Finite splittable groups, Nauka, Moscow, 1968 (in Russian) [34] A. U. Ol’shanskii, Geometry of defining relations in groups, Kluwer Academic Publishers, 1991 [35] E. Jabara, D. V. Lytkina, V. D. Mazurov, “On groups of exponent 72”, J. Group Theory, submitted · Zbl 1323.20032 [36] A. I. Sozutov, “On the structure of the non-invariant factor in some Frobenius groups”, Siberian Math. J., 35:4 (1994), 795-801 · Zbl 0851.20039 [37] V. D. Mazurov, “Groups of exponent 24”, Algebra and Logic, 49:6 (2010), 515-525 · Zbl 1225.20034 [38] A. Kh. Zhurtov, V. D. Mazurov, “Local finiteness of some groups with given element orders”, Vladikavkaz Mat. Zh., 11:4 (2009), 11-15 (in Russian) · Zbl 1324.20024 [39] E. Jabara, D. V. Lytkina, “On groups of period 36”, Siberian Math. J., 54:1 (2013), 29-32 · Zbl 1285.20036 [40] V. D. Mazurov, “Infinite groups with Abelian centralizers of involutions”, Algebra and Logic, 39:1 (2000), 42-49 · Zbl 0960.20025 [41] N. D. Gupta, V. D. Mazurov, “On groups with small orders of elements”, Bull. Austral. Math. Soc., 60 (1999), 197-205 · Zbl 0939.20043 [42] V. D. Mazurov, “Groups of exponent 60 with prescribed orders of elements”, Algebra and Logic, 39:3 (2000), 189-198 · Zbl 0979.20037 [43] V. D. Mazurov, A. S. Mamontov, “On periodic groups with small orders of elements”, Siberian Math. J., 50:2 (2009), 316-321 · Zbl 1212.20068 [44] A. S. Mamontov, “Groups of exponent 12 without elements of order 12”, Siberian Math. J., 54:1 (2013), 114-118 · Zbl 1273.20032 [45] E. Jabara, D. V. Lytkina, A. S. Mamontov, V. D. Mazurov, Groups of period 60, in preparation · Zbl 1327.20043 [46] D. V. Lytkina, A. A. Kuznetsov, “Recognizability by spectrum of the group $$L_2(7)$$ in the class of all groups”, Sib. Electronic Math. Reports, 4 (2007), 300-303 · Zbl 1134.20040 [47] E. Jabara, D. V. Lytkina, A. S. Mamontov, “Recognizing $$M_{10}$$ by spectrum in the class of all groups”, Intern. J. on algebra and computations, 2014 (to appear) · Zbl 1305.20046 [48] D. V. Lytkina, V. D. Mazurov, A. S. Mamontov, “On local finiteness of some groups of period 12”, Siberian Math. J., 53:6 (2012), 1105-1109 · Zbl 1261.20038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.