It is well known that a nontrivial commutator in a free group is never a proper power. We prove a theorem that generalizes this fact and has several worthwhile corollaries. For example, an equation [x1, y1] . . . [xk, yk] = zn, where n ≥ 2k, in the free product F of groups without nontrivial elements of order ≤ n implies that z is conjugate to an element of a free factor of F. If a nontrivial commutator in a free group factors into a product of elements which are conjugate to each other then all these elements are distinct.
- 57M07 (primary)
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