# 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.