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A051576 Order of Burnside group B(3,n) of exponent 3 and rank n. 4
1, 3, 27, 2187, 4782969, 847288609443, 36472996377170786403, 1144561273430837494885949696427, 78551672112789411833022577315290546060373041, 35370553733215749514562618584237555997034634776827523327290883 (list; graph; refs; listen; history; text; internal format)



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.


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.

S. V. Ivanov, On the Burnside problem for groups of even exponent, Proc. Internat. Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 67-75.

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.


Vincenzo Librandi, Table of n, a(n) for n = 0..23

M. Hall, Solution of the Burnside Problem for Exponent Six, Ill. J. Math. 2, 764-786, 1958.

R. C. Lyndon, On Burnside's problem, Transactions of the American Mathematical Society 77, (1954) 202-215.

Todd D. Mateer, A Calculation of an Upper Bound for the Diameter of the Cayley Graph of the Restricted Burnside Group R(2,5)

O'Brien, E. and Newman, M. F. "Application of Computers to Questions Like Those of Burnside, II." Internat. J. Algebra Comput. 6, 593-605, 1996.

J. J. O'Connor and E. F. Robertson, History of the Burnside Problem.

D. Rusin, Burnside Problem. [Broken link?]

D. Rusin, Burnside problem [Cached copy]

Eric Weisstein's World of Mathematics, Burnside Problem


a(n) = 3^(n*(n^2+5)/6) for n >= 0.


3^Table[n*(n^2 + 5)/6, {n, 0, 10}] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)


(Maxima) A051576(n):=3^(n*(n^2+5)/6)$ makelist(A051576(n), n, 0, 7); /* Martin Ettl, Jan 08 2013 */


Equals 3^A004006(n).

Sequence in context: A192341 A191511 A102580 * A184278 A158113 A078233

Adjacent sequences:  A051573 A051574 A051575 * A051577 A051578 A051579




N. J. A. Sloane


Entry revised by N. J. A. Sloane, Jan 12 2016 and Jan 15 2016



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Last modified September 20 16:44 EDT 2019. Contains 327242 sequences. (Running on oeis4.)