A079682 - OEIS

%I #36 Mar 18 2020 18:48:28

%S 1,4,4096,590295810358705651712

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

%C 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]

%C 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.

%C 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.

%C It is not known whether B(5,2) is finite or infinite.

%C See A051576 for additional references.

%D 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.

%D Burnside, William. "On an unsettled question in the theory of discontinuous groups." Quart. J. Pure Appl. Math 33.2 (1902): 230-238.

%D M. Hall, Jr., The Theory of Groups, Macmillan, 1959, Chap. 18.

%D 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.

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

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

%H N. J. A. Sloane, <a href="/A079682/b079682.txt">Table of n, a(n) for n = 0..5</a>

%H S. V. Ivanov, <a href="https://elibm.org/article/10011642">On the Burnside problem for groups of even exponent</a>, Proc. Internat. Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 67-75.

%H E. A. O'Brien and M. F. Newman, <a href="https://www.math.auckland.ac.nz/~obrien/research/burnside.pdf">Application of Computers to Questions Like Those of Burnside, II</a>, Internat. J. Algebra Comput.6, 593-605, 1996.

%H J. J. O'Connor and E. F. Robertson, <a href="http://mathshistory.st-andrews.ac.uk/HistTopics/Burnside_problem.html">History of the Burnside Problem</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BurnsideProblem.html">Burnside Problem</a>

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

%Y Cf. A051576, A004006, A116398, A079682.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Jan 31 2003

%E Entry revised by _N. J. A. Sloane_, Jan 12 2016 and Jan 15 2016