3`,. c2~22,3 ~ y23), where a2i, b2i, c2i E {0,1}, for 1 < i < n, a 2 1 . . b2~ = ~(b2), 4. Consider a3yl vy2wb3 ~ Yl vy2wyzc3 cz)] G simulates this step as l b31. . b3~2a3a . . a3~2yl vy2w3 =~z~ft yl vy2wy33 according to (1 --~ 3`, b3a -+ A , . . , b3~ ---* 3`, 2 ---* 3,, a31 --+ 3 ` , . . , a3~ --* 3`, 2 --~ A, 3 --* y33), where a3~,bai, c3i E {0, t}, for 1 < i < n, a a l .

Hence, for an abelian group K, s = s iff all elements of K have finite order. This is not necessarily true for non-abelian groups. We can, however, prove a pumping lemma which is very similar to the pumping lemma for r e g u l a r languages. L e m m a 1. Let K be some group without elements of infinite order. For any language L C s there is a constant n >_ 1 such that, for all x E L, Ix] >_ n, there exist a decomposition x = uvw and a natural number q > 1 with [~1 < ,~, I~l >- L ~ + ~ c s for ~ll ~ > o.