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A051576
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Order of Burnside group B(3,n) of exponent 3 and rank n.
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4
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1, 3, 27, 2187, 4782969, 847288609443, 36472996377170786403, 1144561273430837494885949696427, 78551672112789411833022577315290546060373041, 35370553733215749514562618584237555997034634776827523327290883
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OFFSET
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0,2
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COMMENTS
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The Burnside group of exponent e and rank r is B(e,r) := F_r / N where F_r is the free group generated by x_1, ..., x_r and N is the normal subgroup generated by all z^e with z in F_r. The Burnside problem is to determine when B(e,r) is finite. [Warning: Some authors interchange the order of e and r. But the symbol is not symmetric. B(i,j) != B(j,i). - N. J. A. Sloane, Jan 12 2016]
B(1,r), B(2,r), B(3,r), B(4,r) and B(6,r) are all finite: |B(1,r)| = 1, |B(2,r)| = 2^r, |B(3,r)| = A051576, |B(4,r)| = A079682, |B(6,r)| = A079683. |B(5,2)| = 5^34.
Many cases are known where B(e,r) is infinite (see references). Ivanov showed in 1994 that B(e,r) is infinite if r>1, e >= 2^48 and 2^9 divides e if e is even.
It is not known whether B(5,2) is finite or infinite.
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REFERENCES
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Burnside, William. "On an unsettled question in the theory of discontinuous groups." Quart. J. Pure Appl. Math 33.2 (1902): 230-238.
M. Hall, Jr., The Theory of Groups, Macmillan, 1959, Chap. 18.
Havas, G. and Newman, M. F. "Application of Computers to Questions Like Those of Burnside." In Burnside Groups. Proceedings of a Workshop held at the University of Bielefeld, Bielefeld, June-July 1977. New York: Springer-Verlag, pp. 211-230, 1980.
Ivanov, Sergei V. "The free Burnside groups of sufficiently large exponents." International Journal of Algebra and Computation 4.01n02 (1994): 1-308. See Math. Rev. MR 1283947.
W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Wiley, 1966, see p. 380.
Novikov, P. S. and Adjan, S. I. "Infinite Periodic Groups I, II, III." Izv. Akad. Nauk SSSR Ser. Mat. 32, 212-244, 251-524, and 709-731, 1968.
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LINKS
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R. C. Lyndon, On Burnside's problem, Transactions of the American Mathematical Society 77, (1954) 202-215.
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FORMULA
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a(n) = 3^(n*(n^2+5)/6) for n >= 0.
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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