cp's OEIS Frontend

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

%I A051576 #55 Feb 16 2025 08:32:41
%S A051576 1,3,27,2187,4782969,847288609443,36472996377170786403,
%T A051576 1144561273430837494885949696427,
%U A051576 78551672112789411833022577315290546060373041,35370553733215749514562618584237555997034634776827523327290883
%N A051576 Order of Burnside group B(3,n) of exponent 3 and rank n.
%C A051576 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 A051576 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 A051576 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 A051576 It is not known whether B(5,2) is finite or infinite.
%D A051576 Burnside, William. "On an unsettled question in the theory of discontinuous groups." Quart. J. Pure Appl. Math 33.2 (1902): 230-238.
%D A051576 M. Hall, Jr., The Theory of Groups, Macmillan, 1959, Chap. 18.
%D A051576 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 A051576 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.
%D A051576 W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Wiley, 1966, see p. 380.
%D A051576 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.
%H A051576 Vincenzo Librandi, <a href="/A051576/b051576.txt">Table of n, a(n) for n = 0..23</a>
%H A051576 M. Hall, <a href="http://projecteuclid.org/euclid.ijm/1255448339">Solution of the Burnside Problem for Exponent Six</a>, Ill. J. Math. 2, 764-786, 1958.
%H A051576 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 A051576 R. C. Lyndon, <a href="https://doi.org/10.1090/S0002-9947-1954-0064049-X">On Burnside's problem</a>, Transactions of the American Mathematical Society 77, (1954) 202-215.
%H A051576 Todd D. Mateer, <a href="http://dimacs.rutgers.edu/REU/1996/sims.html"> A Calculation of an Upper Bound for the Diameter of the Cayley Graph of the Restricted Burnside Group R(2,5)</a>
%H A051576 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 A051576 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 A051576 D. Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/99/burnside">Burnside Problem</a>. [Broken link?]
%H A051576 D. Rusin, <a href="/A051576/a051576.txt">Burnside problem</a> [Cached copy]
%H A051576 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BurnsideProblem.html">Burnside Problem</a>
%F A051576 a(n) = 3^(n*(n^2+5)/6) for n >= 0.
%t A051576 3^Table[n*(n^2 + 5)/6, {n, 0, 10}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 08 2012 *)
%o A051576 (Maxima) A051576(n):=3^(n*(n^2+5)/6)$ makelist(A051576(n),n,0,7); /* _Martin Ettl_, Jan 08 2013 */
%Y A051576 Equals 3^A004006(n).
%K A051576 nonn,easy,nice
%O A051576 0,2
%A A051576 _N. J. A. Sloane_
%E A051576 Entry revised by _N. J. A. Sloane_, Jan 12 2016 and Jan 15 2016