Hostname: page-component-7dd5485656-g8tfn Total loading time: 0 Render date: 2025-10-31T11:54:27.287Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounds in the restricted Burnside problem

Part of: Representation theory of groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Michael Vaughan-Lee  and
E. I. Zel'manov
Michael Vaughan-Lee
Affiliation:
Christ Church Oxford, OX1 1DP England URL: http://users.ox.ac.uk/~vlee/ e-mail: vlee@maths.ox.ac.uk
E. I. Zel'manov
Affiliation:
Department of Mathematics PO Box 208283 10 Hillhouse Avenue New Haven CT 06520-8283 USA e-mail: zelmanov@pascal.math.yale.edu.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We survey the current state of knowledge of bounds in the restricted Burnside problem. We make two conjectures which are related to the theory of PI-algebras.

Keywords

Burnside groupsBurnside problemrestricted Burnside problem.

MSC classification

Secondary: 20F05: Generators, relations, and presentations 20D15: Nilpotent groups, $p$-groups

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 2 , October 1999 , pp. 261 - 271
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Ackermann, W., ‘Zum Hilbertschen Aufbau der reellen Zahlen’, Math. Ann. 99 (1928), 118–133.CrossRefGoogle Scholar
[2]Adjan, S. I., The Burnside problem and identities in groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 95 (Springer, Berlin, 1979).Google Scholar
[3]Adjan, S. I. and Razborov, A. A., ‘Periodic groups and Lie algebras’, Uspekhi Mat. Nauk 42 (1987), 3–68.Google Scholar
[4]Bachmuth, S., Mochizuki, H. Y. and Walkup, D., ‘A nonsolvable group of exponent 5’, Bull. Amer Math. Soc. 76 (1970), 638–640.CrossRefGoogle Scholar
[5]Bahturin, Yu. A., Identical relations in Lie algebras (VNU Science Press BV, 1987).Google Scholar
[6]Bayes, A. J., Kautsky, J. and Wamsley, J. W., Computation in nilpotent groups (application), Lecture Notes in Math. 372 (Springer, Berlin, 1974) pp. 82–89.Google Scholar
[7]Burnside, W., ‘On an unsettled question in the theory of discontinuous groups’, Quart. J. Pure Appl. Math. 33 (1903), 230–238.Google Scholar
[8]Golod, E. S., ‘On nil-algebras and residually finite p-groups’, Izv. Akad. Nauk SSSR, Ser Mat. 28 (1964), 273–276.Google Scholar
[9]Gorenstein, D., Finite simple groups (Plenum Press, New York, 1982).CrossRefGoogle Scholar
[10]Gowers, W. T., personal communication.Google Scholar
[11]Graham, R. L., Rothschild, B. L. and Spencer, J. H., Ramsey theory, Ser. in Discrete Math. (Wiley Interscience, New York, 1990).Google Scholar
[12]Grunewald, F. J., Havas, G., Mennicke, J. L. and Newman, M. F., Groups of exponent eight, Lecture Notes in Math. 806 (Springer, Berlin, 1981) pp. 49–188.Google Scholar
[13]Gupta, N. D. and Newman, M.F., The nilpotency class of finitely generated groups of exponent four, Lecture Notes in Math. 372 (Springer, Berlin, 1974) pp. 330–332.Google Scholar
[14]Hall, M., ‘Solution of the Burnside problem for exponent six’, Illinois J. Math. 2 (1958), 764–786.CrossRefGoogle Scholar
[15]Hall, P. and Higman, G., ‘On the p-length of p-soluble groups and reduction theorems for Burnside's problem’, Proc. London Math. Soc. 6 (1956), 1–42.CrossRefGoogle Scholar
[16]Havas, G., Newman, M. F. and Vaughan-Lee, M. R., ‘A nilpotent quotient algorithm for graded Lie rings’, J. Symbolic Computation 9 (1990), 653–664.CrossRefGoogle Scholar
[17]Havas, G. and Newman, M. F., Applications of computers to questions like those of Burnside, Lecture Notes in Math. 806 (Springer, Berlin, 1980) pp. 211–230.Google Scholar
[18]Havas, G., Wall, G. E. and Wamsley, J. W., ‘The two generator restricted Burnside group of exponent five’, Bull. Austral. Math. Soc. 10 (1974), 459–470.CrossRefGoogle Scholar
[19]Higman, G., ‘On finite groups of exponent five’, Proc. Camb. Phil. Soc. 52 (1956), 381–390.CrossRefGoogle Scholar
[20]Ivanov, S. V., ‘The free Burnside groups of sufficiently large exponent’, Internat. J. Algebra and Comput. 4 (1994), 1–308.CrossRefGoogle Scholar
[21]Kostrikin, A. I., ‘The Burnside problem’, Izv. Akad. Nauk SSSR, Ser Mat. 23 (1959), 3–34.Google Scholar
[22]Kostrikin, A. I., ‘Sandwiches in Lie algebras’, Mat. Sb. 110 (1979), 3–12.Google Scholar
[23]Kostrikin, A. I., Around Burnside, Ergebnisse der Mathematik und ihrer Grenzgebiete (Springer, Berlin, 1990).CrossRefGoogle Scholar
[24]Latyshev, V. N., ‘V. N. Regev's theorem on identities of tensor products of PI-algebras’, Uspekhi Mat. Nauk 27 (1972), 213–214.Google Scholar
[25]Levi, F. and Van der Waerden, B. L., ‘Über eine besondere Kiasse von Gruppen’, Abh. Math. Sem. Univ. Hamburg 9 (1933), 154–158.CrossRefGoogle Scholar
[26]Lysenok, I. G., ‘Infinite Burnside groups of even period’, Izv. Ross. Akad. Nauk Ser. Mat. 60 (1996), 3–224.Google Scholar
[27]Macdonald, I. D., ‘A computer application to finite p-groups’, J. Austral. Math. Soc. Ser. A 17 (1974), 102–112.CrossRefGoogle Scholar
[28]Mann, A. J., ‘On the orders of groups of exponent four’, J. London Math. Soc. 26 (1982), 64–76.CrossRefGoogle Scholar
[29]Neumann, H., Varieties of groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 37 (Springer, Berlin, 1967).CrossRefGoogle Scholar
[30]Newman, M. F. and O'Brien, E. A., ‘Applications of computers to questions like those of Burnside, II’, Internat. J. Algebra and Comput. 6 (1996), 593–605.CrossRefGoogle Scholar
[31]Newman, M. F., ‘Groups of exponent 8 are different’, Bull. London Math. Soc. 25 (1993), 263–264.CrossRefGoogle Scholar
[32]Newman, M. F. and Vaughan-Lee, M., ‘Some Lie rings associated with Burnside groups’, ERA Amer Math. Soc. 4 (1998), 1–3.Google Scholar
[33]Novikov, P. S. and Adjan, S. I., ‘Infinite periodic groups I’, Izv. Akad. Nauk SSSR, Ser. Mat. 32 (1968), 212–244.Google Scholar
[34]Novikov, P. S. and Adjan, S. I., ‘Infinite periodic groups II’, Izv. Akad. Nauk SSSR, Ser. Mat. 32 (1968), 251–524.Google Scholar
[35]Novikov, P. S. and Adjan, S. I., ‘Infinite periodic groups III’, Izv. Akad. Nauk SSSR, Ser Mat. 32 (1968), 709–731.Google Scholar
[36]Razmyslov, Ju. P., ‘On a problem of Hall-Higman’, Izv. Akad. Nauk SSSR, Ser Mat. 42 (1978), 833–847.Google Scholar
[37]Sanov, I. N., ‘Solution of Burnside's problem for exponent four’, Leningrad State Univ. Ann. Math. Ser. 10 (1940), 166–170.Google Scholar
[38]Sims, C. C., Computation with finitely presented groups (Cambridge University Press, Cambridge, 1994).CrossRefGoogle Scholar
[39]Vaughan-Lee, M. R., ‘Lie rings of groups of prime exponent’, J. Austral. Math. Soc. Ser A 49 (1990), 386–398.CrossRefGoogle Scholar
[40]Vaughan-Lee, M. R., The restricted Burnside problem, second edition (Oxford University Press, Oxford, 1993).CrossRefGoogle Scholar
[41]Vaughan-Lee, M. R., ‘The nilpotency class of finite groups of exponent p’, Trans. Amer Math. Soc. 346 (1994), 617–640.Google Scholar
[42]Vaughan-Lee, M. R. and Zel'manov, E. I., ‘Upper bounds in the restricted Burnside problem’, J. Algebra 162 (1993), 107–145.CrossRefGoogle Scholar
[43]Valughan-Lee, M. R., ‘Upper bounds in the restricted Bumside problem II’, Internat. J. Algebra and Comput. 6 (1996), 735–744.CrossRefGoogle Scholar
[44]Wall, G. E., ‘On the Lie ring of a group of prime exponent II’, Bull. Austral. Math. Soc. 19 (1978), 11–28.CrossRefGoogle Scholar
[45]Zel'manov, E. I., ‘The solution of the restricted Burnside problem for groups of odd exponent’, Izv. Math. USSR 36 (1991), 41–60.CrossRefGoogle Scholar
[46]Zel'manov, E. I., ‘The solution of the restricted Burnside problem for 2-groups’, Mat. Sb. 182 (1991), 568–592.Google Scholar
[47]Zel'manov, E. I., ‘On additional laws in the Burnside problem on periodic groups’, Internat. J. Algebra and Comput. 3 (1993), 583–609.CrossRefGoogle Scholar