# Group action

## Definition

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

• ${\displaystyle \forall x.\rho (i,x)=x}$
• ${\displaystyle \forall g,h,x.\rho (g\times _{G}h,x)=\rho (g,\rho (h,x))}$

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

${\displaystyle \rho :G\rightarrow S_{X}}$

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 .

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

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 ${\displaystyle \rho :G\rightarrow \mathrm {Aut} (X)}$ acts transitively iff for any two elements x  and y  in X  there is a group element g  such that ${\displaystyle \rho (g,x)=y}$.

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

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:

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

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

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

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.