Polynomial-time recognizable group

Definition with symbols
Let $$G$$ be a group acting faithfully on a set $$S$$ of size $$n$$. Suppose further that we are given a $$p$$ (polynomial in $$n$$) that takes as input a permutation $$\sigma$$ on $$n$$ elements and outputs whether or not $$\sigma \in G$$.

Then $$p$$ is a polynomial-time algorithm for the membership testing problem in $$G$$, and the pair $$(G,p)$$ is termed a polynomial-time recognizable group.

Note that being polynomial-time recognizable is not a property of the group itself nor is it merely a property of the property action. Rather it describes what we understand of the group. From the fact that membership testing problem can always be solved if we describe a generating set, and the fact that small generating sets exist, it is clear that we can always give a polynomial-time recognition algorithm.

Intersection
An intersection of finitely many polynomial-time recognizable groups is polynomial-time recognizable. Here, what we mean is that given a family of polynomial-time recognizable groups on the same set, say $$(G_1,p_1), (G_2,p_2), ..., (G_r,p_r)$$ there is an algorithm to test membership in the intersection of the $$G_i$$s, which only makes black-box calls to the $$p_i$$s.

Such an algorithm is clear: to test for membership in the intersection, it is necessary and sufficient to test for membership in each member. Thus, the polynomial-time algorithm for the intersection tests that each $$p_i$$ is true.