 Power-map criterion for a finite group

For a finite group $G$, the map $x\mapsto x^m$ is bijective exactly when $m$ is coprime to $|G|$. If the two numbers are coprime, an inverse exponent modulo $|G|$ gives an inverse map. If a prime $p$ divides both, the Cauchy theorem for groups supplies a nonidentity element sent to the identity.