A051576 Order of Burnside group B(3,n) of exponent 3 and rank n.

1, 3, 27, 2187, 4782969, 847288609443, 36472996377170786403, 1144561273430837494885949696427, 78551672112789411833022577315290546060373041, 35370553733215749514562618584237555997034634776827523327290883

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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.

References

• 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.

Links

• 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.
• 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.
• 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)
• E. A. O'Brien and M. F. Newman, 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

Crossrefs

Equals 3^A004006(n).

Programs

Mathematica

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 */

Formula

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

Extensions

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