Generating-set checking reduces to membership testing

Membership testing problem (co-nondeterministic)
Recall that in the membership testing problem, we are given a subset $$A$$ of $$G$$ and are required to formulate a test that will take in an element of $$G$$ and output whether that element is in the subgroup $$H$$ generated by $$A$$.

The generating set-checking problem has a co-nondeterministic reduction to the membership testing problem in the following sense. If a given subset does not generate the whole group, then we can find a short proof of the fact invoking the membership testing problem. Namely, pick an element $$g \in G$$ that is not in the subgroup generated by the $$A$$, and then prove that it actually is not in the subset by invoking the membership testing problem.

Membership testing problem (given a generating set)
If we can find another set $$B$$ that is also a generating set for the group, then the problem of checking whether $$A$$ is a generating set can be reduced to the membership testing problem as follows: for every element of $$B$$, apply the membership testing problem to check whether that element lies in the subgroup generated by $$A$$.

Thus, we have a polynomial-time positive truth table reduction to the membership testing problem provided we already have a generating set.

Membership testing problem (given a random sampler)
If we have a random sampler (that is, an algorithm that saples randomly from the group) we could try the following approach:


 * Use the random sampler to output $$g \in G$$
 * Check whether $$g$$ is in the subgroup generated by $$A$$

Note that if in any such run we get a no, we are confirmed that $$A$$ is not a generating set. If, on the other hand, we get a yes at any stage, we still have the possibility that $$A$$ is a generating set.

If $$H$$ is a proper subgroup of $$G$$, then the index of $$H$$ in $$G$$ is at least 2, hence there are at least as many elements outside $$H$$ as inside $$H$$. Hence, if the random sampler is uniform or near-uniform we have a non-negligible property of hitting on an element outside the group fairly soon.

This thus gives us a co-randomized reduction to the membership testing problem.