A079682 -id:A079682 - cp's OEIS frontend

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

N. J. A. Sloane

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.
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.
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

Equals 3^A004006(n).

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

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

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

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

A079683 Order of Burnside group B(6,n) of exponent 6 and rank n.

Original entry on oeis.org

1, 6, 227442304239437611008

Offset: 0

N. J. A. Sloane, Jan 31 2003

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.
The next term, a(3), is 2^4375*3^833. - N. J. A. Sloane, Jan 12 2016
See A051576 for additional references.

M. Hall, Jr., The Theory of Groups, Macmillan, 1959, Chap. 18.
W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Wiley, 1966, see p. 380.

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

Cf. A051576, A004006, A079682, A116398.

B6n:=proc(n) local a,b,c;
b:=1+(n-1)*2^n;
c:=n+binomial(n,2)+binomial(n,3);
a:=1+(n-1)*3^c;
2^a*3^(b+binomial(b,2)+binomial(b,3));
end; # N. J. A. Sloane, Jan 12 2016

The formula for a(n) was found by Marshall Hall, Jr.: a(n) = 2^i 3^(j + (j choose 2) + (j choose 3)) where i = 1 + (n-1)3^(n + (n choose 2) + (n choose 3)) and j = 1 + (n-1)2^n. (See also the Maple code.)

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

cp's OEIS Frontend

A116398 a(n) = log_2(B[4,n]), where B[4,n] is the order of Burnside group of exponent 4 and rank n (A079682).

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

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A079683 Order of Burnside group B(6,n) of exponent 6 and rank n.

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