For example, obviously all the integer points in a $\mathbb{R}^n$ space have a minimum generating set with a size of $n$, that is, $\{(1,0,...,0),(0,1,...,0),...,(0,0,...,1)\}$.
I came across this because I was thinking what should be the generating set of the Symmetric Group $S_4$, and I thought $\{(12),(23),(34)\}$ would be reasonable, and it was shocking when I realized a smaller set $\{(12),(1234)\}$ can do the job. So is there any way I can find the minimum size of a generating set of a group? Or can I easily tell if my former guessing is wrong?