Clearly every finite group has a minimal generating subset. 


1. Is there any formula for the number of minimal generating subsets of a finite group?
2. Is it known which groups have a unique minimal generating subset?



the number of minimal generating subsets of a group
1.No, this is a hard question in general. It could maybe be done for special classes of groups, say nilpotent groups.


2.The only (finitely-generated) groups which have a unique minimal generating subset are the trivial group and the cyclic group of order 2.


Let $G$ be a group with a unique minimal generating subset $S$. As Gerhard Paseman said in the comments, we can replace a non-involution by its inverse, so we can assume that every element in $S$ is an involution. Now, let $s$ and $t$ be distinct elements of $S$ and let $S^\*=(S\setminus\{t\})\cup \{st\}$. Clearly, $S^\*$ generates $G$. Since $G$ has a unique minimal generating set and $|S|=|S^\*|$, $S^\*$ must be minimal (otherwise we'd get a smaller generating set) and thus $st$ is an involution and $s$ and $t$ commute. Since $s$ and $t$ were arbitrary elements of $S$, $G$ is an elementary abelian $2$-group and it is easily seen that it must have order at most $2$.


