A079682 Order of Burnside group B(4,n) of exponent 4 and rank n.

1, 4, 4096, 590295810358705651712

Offset

0,2

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.

See A051576 for additional references.

References

Bayes, A. J.; Kautsky, J.; and Wamsley, J. W. "Computation in Nilpotent Groups (Application)." In Proceedings of the Second International Conference on the Theory of Groups. Held at the Australian National University, Canberra, August 13-24, 1973(Ed. M. F. Newman). New York: Springer-Verlag, pp. 82-89, 1974.

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.

W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Wiley, 1966, see p. 380.

Tobin, J. J. On Groups with Exponent 4. Thesis. Manchester, England: University of Manchester, 1954.

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..5

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.

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

Eric Weisstein's World of Mathematics, Burnside Problem

Formula

The first few terms are 2 to the powers 0, 2, 12, 69, 422, 2728, that is, 2^A116398(n).

Crossrefs

Cf. A051576, A004006, A116398, A079682.

Sequence in context: A320860 A067482 A249804 * A127235 A274972 A344670

Adjacent sequences: A079679 A079680 A079681 * A079683 A079684 A079685

Keyword

nonn

Author

N. J. A. Sloane, Jan 31 2003

Extensions

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

Status

approved