# Group actions

A group action (AKA left action, realization, representation) of any group $${G}$$ on any set $${X}$$ is a homomorphism $${\rho\colon G\to\textrm{Aut}\left(X\right)}$$, where $${\textrm{Aut}\left(X\right)}$$ is the group of automorphisms of $${X}$$ (the symmetry group). Since in most applications a group action is fixed, we will write the action of $${g\in G}$$ on $${x\in X}$$ as simply $${g(x)}$$, or $${g_{\rho}\left(x\right)}$$ if the homomorphism needs to be made explicit; other common notations are $${gx}$$, $${\rho_{g}(x)}$$ and $${\rho(g)(x)}$$. When acted on by $${G}$$, $${X}$$ is sometimes called a G-set (or a G-space if it is a space).

 Δ As we will see, some specific types of actions (e.g. reps) may be required to preserve additional structures that exist on $${X}$$ (e.g. a vector space structure), but in general, the automorphism group is that of $${X}$$ as a set or space, with any additional structure (e.g. that of a fiber bundle) disregarded.

Since a left action is a homomorphism, $${g\circ h}$$ and $${gh}$$ are required to be the same automorphism, i.e.

$$\displaystyle g\left(h\left(x\right)\right)=\left(gh\right)\left(x\right)\:\forall x\in X.$$

A right action operates from the right within the group, and so instead requires that

$$\displaystyle g\left(h\left(x\right)\right)=\left(hg\right)\left(x\right),$$

and is often written $${xg}$$. Note that a left action can be turned into a right action (and vice versa) via the inverse; e.g. if $${G}$$ has a left action and we define a new action $${g_{R}(x)\equiv g^{-1}(x)}$$, then

\displaystyle \begin{aligned}g_{R}\left(h_{R}\left(x\right)\right) & =g^{-1}\left(h^{-1}(x)\right) \\ & =(g^{-1}h^{-1})(x) \\ & =(hg)^{-1}(x) \\ & =\left(hg\right)_{R}\left(x\right). \end{aligned}

Some definitions related to a group action are:

• Orbit of $${x\in X}$$: $${\textrm{orbit}(x)\equiv\left\{ g(x)\mid g\in G\right\} }$$; i.e. all points of $${X}$$ that can be reached from $${x}$$ by the action of some $${g}$$
• Isotropy group (AKA little group, stabilizer subgroup) of $${x}$$: the subgroup $${I\left(x\right)\equiv\left\{ g\mid g\left(x\right)=x\right\} }$$; i.e. all elements of $${G}$$ that leave $${x}$$ fixed
• Transitive action: $${\forall x,y\:\exists g\mid y=g\left(x\right)\Leftrightarrow X}$$ is a single orbit; i.e. any two points are related by the action of some $${g}$$
• Faithful (AKA effective) action: $${\forall g\neq h\:\exists x\mid g(x)\neq h(x)\Leftrightarrow\rho}$$ is injective; i.e. every $${g}$$ is mapped to a distinct automorphism
• Free (AKA semiregular, fixed point free) action: $${\forall g\neq h,\: g(x)\neq h(x)\:\forall x\Leftrightarrow}$$ only $${e}$$ has a fixed point; i.e. the orbit of every $${x}$$ is an injective map of $${G}$$
• Regular (AKA simply transitive, sharply transitive) action: $${\forall x,y\:\exists}$$ unique $${g\mid y=g(x)\Leftrightarrow}$$ transitive and free; i.e. any two points are related by the action of one $${g}$$

One can state various relationships between these properties, for example: free implies faithful; free is equivalent to all isotropy groups being trivial; and $${G}$$ acts transitively on any orbit of $${X}$$. If the action of $${G}$$ is transitive, then $${X}$$ is called a homogeneous space for $${G}$$; if the action is also free (i.e. regular), then $${X}$$ is called a principal homogeneous space or G-torsor. A $${G}$$-torsor is isomorphic to $${G}$$ as a set or space, but there is no uniquely defined identity element; it can thus be thought of as a group “with the identity forgotten.” The action of $${G}$$ on itself by left or right multiplication is regular.

The above depicts how the action of the three dimensional rotations $${SO(3)}$$ on $${\mathbb{R}^{3}}$$ is not transitive, since two points at different radii cannot be reached from each other by the action of a rotation; is faithful, since every rotation is a distinct automorphism; but is not free, since every rotation leaves an axial line fixed. The orbit of $${x}$$ is the sphere of the same radius, and the isotropy group of $${x}$$ is the two dimensional rotations around the axis it determines.

If a group $${G}$$ has a left action on two sets $${X}$$ and $${Y}$$, a mapping $${f\colon X\to Y}$$ is called equivariant if

$$\displaystyle f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)$$

for all $${g}$$ and $${x}$$. In other words, an equivariant map is a homomorphism with respect to the group action; it is therefore also sometimes called a G-map or G-homomorphism. This definition has to be modified if we extend it to right actions, where we take advantage of the property $${(gh)^{-1}=h^{-1}g^{-1}}$$ to maintain ordering:

Left action on $${Y}$$Right action on $${Y}$$
Left action on $${X}$${\begin{aligned}f\left(g\left(x\right)\right) & =g\left(f\left(x\right)\right)\\ f(gx) & =gf(x) \end{aligned}}{\begin{aligned}f\left(g\left(x\right)\right) & =g^{-1}\left(f\left(x\right)\right)\\ f(gx) & =f(x)g^{-1} \end{aligned}}
Right action on $${X}$${\begin{aligned}f\left(g\left(x\right)\right) & =g^{-1}\left(f\left(x\right)\right)\\ f(xg) & =g^{-1}f(x) \end{aligned}}{\begin{aligned}f\left(g\left(x\right)\right) & =g\left(f\left(x\right)\right)\\ f(xg) & =f(x)g \end{aligned}}

The equivariance condition for a map $${f\colon X\to Y}$$ between two $${G}$$-sets, using two common notations.

If $${G}$$ has a left action on $${X}$$, and we denote the left cosets of the isotropy group as $${G/I(x)}$$, then the map $${f:G/I(x)\rightarrow\mathrm{orbit}(x)}$$ defined by $${gI(x)\mapsto gx}$$ is equivariant. The orbit-stabilizer theorem states that this map is also bijective. Such a map is sometimes called a $${G}$$-map isomorphism. For finite $${G}$$, the corollary $${\left|G\right|/\left|I(x)\right|=\left|G:I(x)\right|=\left|\mathrm{orbit}(x)\right|}$$ is also sometimes referred to as the orbit-stabilizer theorem, where $${\left|\mathrm{orbit}(x)\right|}$$ denotes the number of elements in the set.

A Lie group has the additional structure of a differentiable manifold, which is required to carry over the action homomorphism to the corresponding automorphisms. Thus a Lie group action is defined to be a smooth homomorphism from a Lie group $${G}$$ to $${\textrm{Diff}(M)}$$, the Lie group of diffeomorphisms of a manifold $${M}$$.