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# Determining if a set is a Generating Set given a known group rank

2017-12-03 03:07:20

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, this is a necessary condition:

$\forall s \in \{s_i\} \; \;\not \exists g \in G \; : \; g \neq s \wedge s \in \langle g \rangle$

Is it also sufficient?