TITLE: How to find the smallest set of generating elements in a group?
QUESTION [5 upvotes]: Is there a systematic procedure for finding the smallest set of generating elements of a finite group?

REPLY [3 votes]: A following method can be used to find the smallest set of the generating elements in a finite group:
It is based on the following theorem:

Suppose $G$ is a finite group, $\{X_i\}_{i = 1}^{n}$ are i.i.d uniformly distributed random elements of $G$. Then $P(\langle \{X_i\}_{i = 1}^{n} \rangle = G) = \sum_{H \leq G} \mu(G, H) {\left(\frac{|H|}{|G|}\right)}^n$, where $\mu$ is the Moebius function for subgroup lattice of $G$.

Thus, the smallest possible cardinality of a generating set can be described by the following formula:
$$\min\{n \in \mathbb{N}| \sum_{H \leq G} \mu(G, H) {\left(\frac{|H|}{|G|}\right)}^n > 0\}$$
And if we know the smallest possible cardinality of a generating set (let's denote it by $s$), then we can find an example by checking all of the $C_{|G|}^s$ subsets on whether they lie in one of the maximal proper subgroups or not. If they lie - they are the smallest possible generating set.