Coset representative function

Setup
Let $$G$$ be a group with an encoding $$C$$. That is, $$C$$ associates to each element of $$G$$ a string over a fixed (say, binary) alphabet, along with algorithms for testing validity of a code-word, for multiplying group elements, and for finding the inverse of a group element.

Let $$H G$$ be a subgroup.

Definition part
A coset representative function for $$H$$ in $$G$$ is a function $$f:G \to G$$ that is constant on the left cosets and that sends each left coset to an element in that left coset.

Another way of viewing this is that a coset representative function for $$H$$ in $$G$$ involves first choosing a system of coset representatives for $$H$$ in $$G$$, and then defining a map that sends each element to the coset representative of its left coset.

Weaker descriptions

 * Coset enumeration
 * Coset-separating function
 * Membership test

Intersection of subgroups
Given subgroups $$H_1$$ and $$H_2$$ of $$G$$ with coset representative functions $$f_1$$ and $$f_2$$, can we obtain a coset representative function for $$H_1 \cap H_2$$?

Composition operator
Suppose we have a coset representative function for $$H$$ in $$K$$, and a coset representative function for $$K$$ in $$G$$. Then, we can use both of them to obtain a coset representative function for $$H$$ in $$G$$, as follows:


 * The set of coset representatives is the set of products $$gh$$ where $$g$$ is a coset representative for $$K$$ in $$G$$, and $$h$$ is a coset representative for $$H$$ in $$K$$.


 * To find the coset representative of an element of $$G$$ with respect to $$H$$, we first compute the coset representative with respect to $$K$$, then take the quotient and compute the coset representative of the quotient with respect to $$H$$. Multiplying the two values out gives the coset representative.

Transfer operator
It is not clear how we can use the coset representative function for $$H$$ in $$G$$ to obtain a coset representative function for $$H \cap K$$ in $$K$$.