# Action-core description

## Definition

### Setup

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

### Definition part

An action-core description of $N$ is a given action of $G$ on a set $X$ such that the kernel of the action is precisely $N$.

More explicitly:

• We are given a set $X$ whose elements are words in some language (over some fixed alphabet) with a membership test for $X$
• We are given an algorithm that. given any $g$ in $G$ and any $x \in X$, takes in the encodings of $g$ and $x$ and outputs the encoding of $g.x$

## Relation with other descriptions

In the general case, the action-core description describes that normal subgroup which is the intersection of all the isotropy subgroups (by this is meant the isotropy subgroups at all points).

In the particular case when we consider the action on the coset space, this is equivalent to a coset enumeration.