A079683 Order of Burnside group B(6,n) of exponent 6 and rank n.

1, 6, 227442304239437611008

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.

The next term, a(3), is 2^4375*3^833. - N. J. A. Sloane, Jan 12 2016

See A051576 for additional references.

References

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

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

Links

Table of n, a(n) for n=0..2.

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.

J. J. O'Connor and E. F. Robertson, History of the Burnside Problem

Formula

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.)

Maple

B6n:=proc(n) local a, b, c;
b:=1+(n-1)*2^n;
c:=n+binomial(n, 2)+binomial(n, 3);
a:=1+(n-1)*3^c;
2^a*3^(b+binomial(b, 2)+binomial(b, 3));
end; # N. J. A. Sloane, Jan 12 2016

Crossrefs

Cf. A051576, A004006, A079682, A116398.

Sequence in context: A072240 A182795 A182796 * A115528 A354995 A270032

Adjacent sequences: A079680 A079681 A079682 * A079684 A079685 A079686

Keyword

nonn,bref

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