0,2

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.

B(1,r), B(2,r), B(3,r), B(4,r) and B(6,r) are all finite: |B(1,r)| = r, |B(2,r)| = 2^r, |B(3,r)| = A051576, |B(4,r)| = A079682, |B(6,r)| = A079683.

B(e,r) is infinite for e > 2 and n >= 13 (Ivanov).

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

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.

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

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

D. Rusin, Burnside Problem

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.

Sequence in context: A045518 A058457 A072240 this_sequence A115528 A106226 A005070

Adjacent sequences: A079680 A079681 A079682 this_sequence A079684 A079685 A079686

nonn,bref

N. J. A. Sloane (njas(AT)research.att.com), Jan 31 2003

The next term is too large to include.