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%I #23 Mar 18 2020 14:51:55
%S 1,6,227442304239437611008
%N Order of Burnside group B(6,n) of exponent 6 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 The next term, a(3), is 2^4375*3^833. - _N. J. A. Sloane_, Jan 12 2016
%C See A051576 for additional references.
%D M. Hall, Jr., The Theory of Groups, Macmillan, 1959, Chap. 18.
%D W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Wiley, 1966, see p. 380.
%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 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>
%F 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.)
%p B6n:=proc(n) local a,b,c;
%p b:=1+(n-1)*2^n;
%p c:=n+binomial(n,2)+binomial(n,3);
%p a:=1+(n-1)*3^c;
%p 2^a*3^(b+binomial(b,2)+binomial(b,3));
%p end; # _N. J. A. Sloane_, Jan 12 2016
%Y Cf. A051576, A004006, A079682, A116398.
%K nonn,bref
%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
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