Sanov's theorem on Lie relators in groups of exponent $p$

Document Type : Research Article

Author

Christ Church, University of Oxford, Oxford, OX1 1DP, England.

Abstract

‎I give a proof of Sanov's theorem that the Lie relators of weight at most‎ ‎$2p-2$ in groups of exponent $p$ are consequences of the identity $px=0$ and‎ ‎the $(p-1)$-Engel identity‎. ‎This implies that the order of the class $2p-2$‎ ‎quotient of the Burnside group $B(m,p)$ is the same as the order of the class‎ ‎$2p-2$ quotient of the free $m$ generator $(p-1)$-Engel Lie algebra over‎ ‎GF$(p)$‎. ‎To make the proof self-contained I have also included a derivation of‎ ‎Hausdorff's formulation of the Baker Campbell Hausdorff formula‎.

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References

S. I. Adjan, The Burnside Problem and Identities in Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 95, Springer-Verlag, Berlin, 1979.
S. Bachmuth, Solution to the Burnside Problem, arXiv:0803.1612, 2008.
S. Bachmuth, A straightforward solution to the Burnside Problem, arXiv:1603.08421, 2016.
H. E. Baker, Alternants and continuous groups, Proc. London Math. Soc. 3 (1905) 24--47.
E. Hausdorff, Die symbolische exponentialformel in der gruppentheorie, Berichte der Saechsischen Akademie der Wissenschaften, (Math.-Phys. Kl.) Leipsig 58 (1906) 19--48.
G. Havas, G. E. Wall and J. W. Wamsley, The two generator restricted Burnside group of exponent five, Bull. Austral. Math. Soc. 10 (1974) 459--470.
G. Havas and M. F. Newman, Applications of computers to questions like those of Burnside, Lecture Notes in Mathematics, 806, Berlin, Springer-Verlag, (1974) 330--332.
G. Higman, Some remarks on varieties of groups, Quart. J. Math. Oxford Ser. (2) 10 (1959) 165--178.
N. Jacobson, Lie Algebras, Wiley-Interscience, New York, 1962.
A. I. Kostrikin, On the connection between periodic groups and Lie rings (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 21 (1957) 289--310.
W. Magnus, A connection between the Baker-Hausdorff formula and a problem of Burnside, Ann. of Math. (2) 52 (1950) 111--126.
W. Magnus, A. Karrass and D. Solitar Combinatorial group theory, Presentations of groups in terms of generators and relations, Second Revised ed., Dover Publications, Inc., New York, 1976.
E. A. O'Brien and M. Vaughan-Lee, The 2-generator restricted Burnside group of exponent 7, Internat. J. Algebra Comput. 12 (2002), no. 4, 575--592.
A. Ju. Olschanskii, Groups of bounded period with subgroups of prime order, Algebra and Logic 21 (1982) 369--418.
I. N. Sanov, Establishment of a connection between periodic groups with period a prime number and Lie rings, Izv. Akad. Nauk SSSR, Ser. Mat. 16 (1952) 23--58.
M. R. Vaughan-Lee, The Restricted Burnside Problem, second ed., Oxford University Press, 1993.
Journal of the Iranian Mathematical Society
Volume 2, Issue 1
January 2021
Pages 1-16

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