Group action

Definition
Classically a group action is a function $$\rho : G \times X \rightarrow X$$ where $G$ is a group with identity $i$ and $X$ is a set such that:


 * $$\forall x. \rho(i, x) = x$$
 * $$\forall g,h,x. \rho(g\times_G h,x)=\rho(g,\rho(h,x))$$

In this way the group action preserves some qualities of the group. These requirements ensure that for every element $g$ in $G$, $$\rho(g)$$ acts as a permutation on $X$. Thus this can equivalently be formulated to say that $$\rho$$ is a homomorphism from $G$ to the group of permutations on $X$.

$$\rho : G \rightarrow S_X$$

From here the definition can be simplified and generalized to: A group action is a group homomorphism from a group $G$ to an automorphism group of some object $X$.

$$\rho : G \rightarrow \mathrm{Aut}(X)$$

The permutation group of a set is its automorphism group, but this definition allows groups to act on more than just sets. For example it is useful to think of a group acting on an abstract polytope, that is a homomorphism from the group to the group of automorphisms on the polytope.

This notion can be extended further. For example despite requiring that $G$ being a group this definition does not use the inverse property and thus works perfectly for monoids as well. And furthermore the closure isn't used so $G$ can be relaxed further to simply be a category. With the definition relaxed this much a group action is equivalent to the definition of a functor.

Transitivity
An important concept for polytopes is the concept of transitivity. A group action $$\rho : G \rightarrow \mathrm{Aut}(X)$$ acts transitively iff for any two elements $x$ and $y$ in $X$ there is a group element $g$ such that $$\rho(g,x)=y$$.

For example the definition of isogonal is that the symmetry group of a polytope acts transitively on its vertices. That is that for any two vertices of the polytope there is a member of the group that maps between them.

Orbit
For a group action $$\rho : G \rightarrow \mathrm{Aut}(X)$$ and an element $$x : X$$, the orbit of $x$ is the set of elements $$\{\rho(g,x)\mid g \in G\}$$.

The group action acts transitively on all orbits, and thus the orbits partition a set.

Every element is in its own orbit and an element whose orbit consists entirely of itself is called a fixed point.

Faithful actions
A faithful action is an action with the additional stipulation that it is an injective homomorphism. This can also be stated as:

$$ \rho(g) = \mathrm{id} \iff g = i $$

That is, if an element of $G$ maps to the identity automorphism, it must be an identity in the group.

Free actions
A free action is an action $$\rho : G \rightarrow \mathrm{Aut}(X)$$ such that:

$$ (\exists x. \rho(g,x) = x) \iff g = i $$

That is, if an element of $G$ fixes an element of $X$, it must be an identity in the group. This is a stronger condition than faithful and all free actions are also faithful.

Examples

 * For a group $G$ its own operator is a an action acting on itself. As a result sometimes the group operator itself is called a "group action".
 * For any group $G$ and object $X$ there is a trivial group action, which always yields the identity automorphism.