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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the Lie algebra of a Burnside group of exponent $ 5$

Authors: Eugene F. Krause and Kenneth W. Weston
Journal: Proc. Amer. Math. Soc. 27 (1971), 463-470
MSC: Primary 17.30; Secondary 20.00
MathSciNet review: 0276287
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Abstract: A. Kostrikin proved that $ R(5,2)$ has order at most 5$ ^{34}$ and is nilpotent of class at most 13. He verified this by showing that the freest Lie ring $ L$ on 2 generators of characteristic 5 which satisfies the 4th Engel condition must be nilpotent of class at most 13. He then appealed to the correspondence between Lie rings and groups, given by the associated Lie ring of a group, to complete his proof. We studied Kostrikin's calculations and considered a collection of matrices based on them. From these, we were able to construct $ L$ with the use of a Univac 1107 computer and verify that $ L$ is nilpotent of class exactly 13 with order exactly 5$ ^{34}$. Hence, modulo the conjecture made by Kostrikin, Sanov and others that $ L$ is the associated Lie ring of $ R(5,2)$, the order of $ R(5,2)$ is exactly 5$ ^{34}$.

References [Enhancements On Off] (What's this?)

  • [1] Marshall Hall Jr., The theory of groups, The Macmillan Co., New York, N.Y., 1959. MR 0103215
  • [2] Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. MR 0143793
  • [3] A. I. Kostrikin, Solution of a weakened problem of Burnside for exponent 5, Izv. Akad. Nauk SSSR. Ser. Mat. 19 (1955), 233–244 (Russian). MR 0071433

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Keywords: Restricted Burnside group of exponent 5, associated Lie ring, freest Lie algebra, generators, GF$ (5)$, 4th Engel condition
Article copyright: © Copyright 1971 American Mathematical Society