Membership test (subgroup description)

Definition part
A membership test for $$H$$ in $$G$$ is a test that takes as input the code-word for some element of $$G$$ (that is, it takes as input a string guaranteed to be a code-word for an element of $$G$$) and outputs whether it is a code-word for an element in $$H$$.

Stronger subgroup description rules
Many subgroup description rules automatically give a membership test. These include:


 * Coset enumeration
 * Coset-separating function
 * Coset representative function

Relation with a generating set
A priori, it is not clear whether a membership test provides us an efficient generating set, nor is it clear whether a generating set provides us with an efficient membership test. However, it turns out that when $$G$$ is the symmetric group equipped with its usual encoding, we can in fact obtain a membership test from a generating set.

The general problem of finding a membership test from a generating set is termed the membership testing problem.

Effect of subgroup operators
We are interested in pairs $$(H,M)$$ where $$H$$ is a subgroup of $$G$$ and $$M$$ is a membership test for $$H$$ in $$G$$.

Intersection of subgroups
Let $$H_1$$ and $$H_2$$ be two subgroups of $$G$$ with membership tests $$M_1$$ and $$M_2$$ respectively. Then, $$H_1 \cap H_2$$ has membership test $$M_1 \land M_2$$.

Suppose $$H$$ has a membership test in $$G$$, and $$K$$ has a membership test in $$H$$. Then we get a natural membership test for $$K$$ in $$G$$.

Transfer operator
A membership test for $$H$$ in $$G$$ also gives a membership test for $$H \cap K$$ in $$K$$.

Related subgroup properties
In general, we are interested in subgroups for which there exist fast algorithms for membership testing. This notion of fast usually means fast as a function of the size of the subgroup, possibly the index of the subgroup, and possibly the logarithm of the size of the whole group.

Polynomial time-recognizability
An encoding of an IAPS of groups is termed subgroup-polynomial-time-recognizable if there exists a polynomial $$p$$ such that the following holds: For any subgroup $$H$$ of the $$n^{th}$$ member, there is an algorithm whose running time is bound above by $$p(n)$$ that acts as a membership test for $$H$$ in $$G_n$$.

An operator on subgroups is termed polynomial-time-recognizability-preserving if the time bound on the membership test on the subgroup obtained after applying the operator, is bounded by a polynomial in the time bounds on the membership test on the original subgroups.

Thus, from observations above, the intersection operator, the transfer operator, the composition operator, are all polynomial-time-recognizability-preserving.