Coset enumeration

This article defines a subgroup description rule, viz a rule that allows us to describe subgroups of a group in terms of encodings of the whole group

Definition

Setup

A group $G$ equipped with an encoding $C$. An abstract subgroup $H$ of $G$.

Definition part

A coset enumeration of $H$ in $G$ is essentially a description of the action of $G$ on the coset space $X=G/H$. It provides the following data:

• A set $X$ encoded as strings, with a membership test for $X$
• An algorithm that takes as input $g\in G$ and $x\in X$ (in terms of their encodings) and outputs $g.x$
• A distinguished element in $X$ for which we are told that the isotropy is exactly $H$

Relation with other description rules

Stronger description rules

• Coset representative function

Weaker description rules

• Coset-separating function
• Membership test