On the Burnside Problem for Groups of Even Exponent

The Burnside problem about periodic groups asks whether any finitely generated group with the law x ≡ 1 is necessarily finite. This is proven only for n ≤ 4 and n = 6. A negative solution to the Burnside problem for odd n ≫ 1 was given by Novikov and Adian. The article presents a discussion of a recent solution of the Burnside problem for even exponents n ≫ 1 and related results. 1991 Mathematics Subject Classification: Primary 20F05, 20F06, 20F10, 20F50 Recall that the notorious Burnside problem about periodic groups (posed in 1902, see [B]) asks whether any finitely generated group that satisfies the law x ≡ 1 (n is a fixed positive integer called the exponent of G) is necessarily finite. A positive solution to this problem is obtained only for n ≤ 4 and n = 6. Note the case n ≤ 2 is obvious, the case n = 3 is due to Burnside [B], n = 4 is due to Sanov [S], and n = 6 to M. Hall [Hl] (see also [MKS]). A negative solution to the Burnside problem for odd exponents was given in 1968 by Novikov and Adian [NA] (see also [Ad]) who constructed infinite m-generator groups with m ≥ 2 of any odd exponent n ≥ 4381 (later Adian [Ad] improved on this estimate bringing it down to odd n ≥ 665). A simpler geometric solution to this problem for odd n > 10 was later given by Ol’shanskii [Ol1] (see also [Ol2]). We remark that attempts to approach the Burnside problem via finite groups gave rise to a restricted version of the Burnside problem [M] that asks whether there exists a number f(m,n) so that the order of any finite m-generator group of exponent n is less than f(m,n). The existence of such a bound f(m,n) was proven for prime n by Kostrikin [K1] (see also [K2]) and for n = p with prime p by Zelmanov [Z1]-[Z2]. By a reduction theorem due to Ph. Hall and Higman [HH] it then follows from this Zelmanov result that, modulo the classification of finite simple groups, the function f(m,n) does exist for all m,n. However, the Burnside problem for even exponents n without odd divisor ≥ 665, being especially interesting for n = 2 ≫ 1, remained open. The principal difference between odd and even exponents in the Burnside problem can be illustrated by pointing out that, on the one hand, for every odd n ≫ 1 there are infinite Supported in part by the Alfred P. Sloan Foundation and the National Science Foundation of the United States Documenta Mathematica · Extra Volume ICM 1998 · II · 67–75