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

Eric Weisstein's World of Mathematics, Burnside Problem

Sequence in context: A024061 A067482 A013830 this_sequence A127235 A102205 A046360

Adjacent sequences: A079679 A079680 A079681 this_sequence A079683 A079684 A079685

nonn

njas, Jan 31 2003

The next term is 2^422.