Groups with given element orders

Daria V. Lytkina ; Victor D. Mazurov

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 7 (2014) no. 2, pp. 191-203.

Résumé

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).

Détail

Keywords: spectrum, exponent, periodic group, locally finite group.

Bibliographie

[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 | DOI | MR | Zbl

[2] M. Hall, “Solution of the Burnside problem for exponent six”, Illinois J. Math., 2 (1958), 764–786 | MR | Zbl

[3] W. Burnside, “On an unsettled question in the theory of discontinuous groups”, Quart. J. Pure Appl. Math., 33 (1902), 230–238 | Zbl

[4] S. I. Adyan, “The Burnside problem and related questions”, Russian Math. Surveys, 65:5 (2010), 805–855 | DOI | MR | Zbl

[5] M. A. Hilton, An introduction to the theory of groups of finite order, Clarendon Press, Oxford, 1908 | Zbl

[6] W. Burnside, “On groups in which every two conjugate operations are permutable”, Proc. London Math. Soc., 35 (1903), 28–37 | Zbl

[7] C. Hopkins, “Finite groups in which conjugate operations are commutative”, Amer. J. Math., 51 (1929), 35–41 | DOI | MR | Zbl

[8] F. Levi, “Groups in which the commutator operations satisfy certain algebraic conditions”, J. Indian Math. Soc., 6 (1942), 166–170 | MR

[9] F. Levi, B. van der Waerden, “Über eine besondere Klasse von Gruppen”, Abh. Math. Semin., Hamburg Univ., 9 (1932), 157–158

[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) | MR | Zbl

[11] Yu. P. Razmyslov, “The Hall-Higman problem”, Math. USSR – Izv., 13:1 (1979), 133–146 | DOI | MR | Zbl

[12] A. I. Kostrikin, Around Burnside, Ergeb. Math. Grenzgeb. (3), 20, Springer-Verlag, New York–Berlin–Heidelberg, 1990 | MR | Zbl

[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 | DOI | MR | Zbl

[14] M. F. Newman, “Groups of exponent six”, Computational group theory (Durham, 1982), Academic Press, London, 1984, 39–41 | MR

[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 | DOI | MR | Zbl

[16] P. S. Novikov, “On periodic groups”, Dokl. Akad. Nauk, 127 (1959), 749–752 (in Russian) | MR | Zbl

[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) | MR | Zbl

[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) | MR | Zbl

[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) | MR | Zbl

[20] S. I. Adian, The Burnside problem and identities in groups, Springer-Verlag, Berlin–New York, 1978 | MR

[21] S. V. Ivanov, “The free Burnside groups of sufficiently large exponents”, Internat. J. Algebra Comput., 4 (1994), 3–308 | DOI | MR

[22] I. G. Lysenok, “Infinite Burnside groups of even exponent”, Izv. Math., 60:3 (1996), 453–654 | DOI | MR | Zbl

[23] B. H. Neumann, “Groups whose elements have bounded orders”, J. London Math. Soc., 12 (1937), 195–198 | DOI | MR | Zbl

[24] D. V. Lytkina, “Structure of a group with elements of order at most 4”, Siberian Math. J., 48:2 (2007), 283–287 | DOI | MR | Zbl

[25] B. H. Neumann, “Groups with automorphisms that leave only the neutral element fixed”, Arch. Math., 7 (1956), 1–5 | DOI | MR | Zbl

[26] A. Kh. Zhurtov, “Regular automorphisms of order 3 and Frobenius pairs”, Siberian Math. J., 41:2 (2000), 268–275 | DOI | MR | Zbl

[27] E. Jabara, “Fixed point free action of groups of exponent 5”, J. Austral. Math. Soc., 77 (2004), 297–304 | DOI | MR | Zbl

[28] E. Jabara, “Free actions of groups of exponent 5”, Algebra and Logic, 50:5 (2011), 466–469 | DOI | MR | Zbl

[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)

[30] E. Jabara, P. Mayr, “Frobenius complements of exponent dividing $2^m\cdot9$”, Forum Mathematicum, 21:1 (2009), 217–220 | MR | Zbl

[31] D. V. Lytkina, “Periodic groups acting freely on abelian groups”, Algebra and Logic, 49:3 (2010), 256–264 | DOI | MR | Zbl

[32] V. P. Shunkov, “On a class of $p$-groups”, Algebra i logika, 9:4 (1970), 484–496 (in Russian) | DOI | MR

[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 | MR

[35] E. Jabara, D. V. Lytkina, V. D. Mazurov, “On groups of exponent 72”, J. Group Theory, submitted

[36] A. I. Sozutov, “On the structure of the non-invariant factor in some Frobenius groups”, Siberian Math. J., 35:4 (1994), 795–801 | DOI | MR | Zbl

[37] V. D. Mazurov, “Groups of exponent 24”, Algebra and Logic, 49:6 (2010), 515–525 | DOI | MR

[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) | MR

[39] E. Jabara, D. V. Lytkina, “On groups of period 36”, Siberian Math. J., 54:1 (2013), 29–32 | DOI | MR | Zbl

[40] V. D. Mazurov, “Infinite groups with Abelian centralizers of involutions”, Algebra and Logic, 39:1 (2000), 42–49 | DOI | MR | Zbl

[41] N. D. Gupta, V. D. Mazurov, “On groups with small orders of elements”, Bull. Austral. Math. Soc., 60 (1999), 197–205 | DOI | MR | Zbl

[42] V. D. Mazurov, “Groups of exponent 60 with prescribed orders of elements”, Algebra and Logic, 39:3 (2000), 189–198 | DOI | MR | Zbl

[43] V. D. Mazurov, A. S. Mamontov, “On periodic groups with small orders of elements”, Siberian Math. J., 50:2 (2009), 316–321 | DOI | MR | Zbl

[44] A. S. Mamontov, “Groups of exponent 12 without elements of order 12”, Siberian Math. J., 54:1 (2013), 114–118 | DOI | MR | Zbl

[45] E. Jabara, D. V. Lytkina, A. S. Mamontov, V. D. Mazurov, Groups of period 60, in preparation

[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 | MR | Zbl

[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) | MR

[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 | DOI | MR | Zbl