Group in which no non-identity element has arbitrarily large roots

Definition
A group in which no non-identity element has arbitrarily large roots is a group $$G$$ with the property that if $$g$$ is a non-identity element of $$G$$, there exists some natural number $$n$$ (dependent on $$g$$) such that the equation $$x^n = g$$ has no solution.

Facts

 * Finitary symmetric group implies no non-identity element has arbitrarily large roots: This is used to prove that locally finite not implies embeddable in finitary symmetric group, because there are locally finite groups that don't have this property and they cannot be embedded in the finitary symmetric group, which has the property.