Suppose I have a finitely generated group $G$ of known rank $n$, and a set $\{s_i\}$ of $n$ group elements. Are there some simple necessary and sufficient conditions to determine whether $s_i$ generates $G$? (Suppose that I don't have any known generating set which I can try to generate with the $\{s_i\}$.)
For example, I think this is a necessary condition:
- $\forall s \in \{s_i\} \; \;\not \exists g \in G \; : \; \langle s\rangle \subset \langle g \rangle$
Is it also sufficient?
The group $\langle a, t; t^{-1}at=a^2\rangle$ has rank two and is generated by the set $\{a^2, t\}$. However, $\langle a^2\rangle\lneq\langle a\rangle$ (this follows from Magnus' Freiheitssats*, as $\langle a\rangle\cong\mathbb{Z}$). Therefore, your condition is neither necessary or sufficient.
I would suspect that in general there does not exist an algorithm to determine, for a group $G$ of rank $n$, if a given $n$-element subset of generates the whole group $G$. However, I cannot come up with a proof at the moment...
*See Magnus, Karrass and Solitar, Combinatorial Group Theory, 1965