The dihedral group has the following presentation $$D_{2i}=\left<s,r\mid s^2=r^i=e,sr=r^{-1}s \right>.$$
The order of $D_{2i}$ is $2i$. If we take $S=\{s,r\}$ and $H=\{s,sr\}$, then $D_{2i}=\left<S \right>=\left<H\right>$, but the sum the orders of elements of $S$ is $2+i$ and of $H$ is $4$. From this we deduce that there is no relation between the order of the non-abelian group and the orders of elements in minimal generating set.
My question is there any relation between the order of abelian group and the order of elements in its minimal generating sets?