Define the order ord$(g)$ of an element $g$ in a finite group $G$ to be $|\{g^0,g^1,g^2,\ldots\}|$.
I want to prove an alternate definition of the order of an element, but I don't know how to prove this implication:
Show for an arbitrary $g\in G$ and $d\in \mathbb{Z}_{>0}$: If $g^d=1$ and $g^{d/t}\neq 1$ for all prime divisors $t$ of $d$, then ord$(g)=d$.