In this article we give a sufficient and necessary condition to determine whether an element of the free group induces a nontrivial element of the free Burnside group of sufficiently large odd exponents. Although this result is “well known” among specialists, it has never been stated with such a level of simplicity. Moreover, our proof highlights some important differences between the Delzant-Gromov approach to the Burnside problems and others that exist. This criterion can be stated without any knowledge regarding Burnside groups, in particular about the proof of its infiniteness. Therefore, it also provides a useful tool to study outer automorphisms of Burnside groups. In addition, we state an analogue result for periodic quotients of torsion-free hyperbolic groups
