On groups with a permutational property on commutators of weight 4

Abstract.

We show that a periodic 2-generator group G with no involutions is metabelian if and only if for any $a, x_1, x_2,x_3 \in G$ there exists a nontrivial permutation $\sigma \in \Bbb {S}_3$ such that $[a, x_1, x_2, x_3 ] = [a,x_{\sigma (1)},x_{\sigma (2)},x_{\sigma (3)}]$.