POWERS OF SUBSETS IN FREE PERIODIC GROUPS

It is proved that for every odd n≥1039 there are two words u(x,y),v(x,y) of length ≤658n2 over the group alphabet {x,y} of the free Burnside group B(2,n), which generate a free Burnside subgroup of the group B(2,n). This implies that for any finite subset S of the group B(m,n) the inequality |St|>4⋅2.9[t658s2] holds, where s is the smallest odd divisor of n that satisfies the inequality s≥1039.