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A079682 Order of Burnside group B(4,n) of exponent 4 and rank n. 4
1, 4, 4096, 590295810358705651712 (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.
See A051576 for additional 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.
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
The first few terms are 2 to the powers 0, 2, 12, 69, 422, 2728, that is, 2^A116398(n).
Sequence in context: A320860 A067482 A249804 * A127235 A274972 A344670
N. J. A. Sloane, Jan 31 2003
Entry revised by N. J. A. Sloane, Jan 12 2016 and Jan 15 2016

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