Group action

Let $X$ be a set and $G$ a group. A right (left) action of $G$ on $X$ is a map

denoted b $(x,g)\mapsto x\cdot g$, and such that $x\cdot 1=x$ y $(x\cdot g)\cdot g^{\prime }=x\cdot (g{g}^{\prime })$ for every $x\in X$ and $g,g^{\prime }\in G$.

One may equivalently define a group action of $G$ on $X$ as a group homomorphism from $G$ into the symmetric group $Sym(X)$ of all bijections from $X$ to itself.

There is also an interpretation in terms of category theory (see group as a category).

Related notions: subaction and quotient action.

Types

• A free action.
• A transitive action.
• A faithful action.
• A regular action.
• A proper action.

More definitions

In the conditions of the previous definition, we call the $G$-orbit of $x\in X$ the set of points $x\cdot g$ for $g\in G$.

For a subset $S\subseteq X$ and an element $g\in G$, the $g$-translate $Sg$ is the set of points in $X$ of the form $x=s\cdot g$, for some $s\in S$.

The quotient set $X/G$ is the set of $G$-orbits, and the map $\pi :X\to X/G$ that sends $x$ to its $G$-orbit is the quotient map.

If the action is transitive, then there is only one orbit, and for any $x\in X$, we have

where $G/{\text{Stab}}_G(x)=\{g\cdot {\text{Stab}}_G(x):g\in G\}$, the set of cosets. In this case, we say that $X$ is a homogeneous space.

Let $X$ be a topological space and $G$ a discrete group. A right action of $G$ on $X$ is continuous if, for each $g\in G$, the induced map $X\to X$ is continuous (and therefore a homeomorphism).

The action is said to be properly discontinuous when it is continuous and each $x\in X$ has a neighborhood $U_x$ such that the $G$-translate $U_x\cdot g$ intersects $U_x$ only for a finite number of elements of $G$.

Free and properly discontinuous actions are important because, in Hausdorff spaces $X$, each $x\in X$ has at least one neighborhood $U_x$ disjoint from each $U_x\cdot g$ for any $g\ne 1$ (this needs to be proven).

Let $X$ be a Hausdorff topological space equipped with a free and properly discontinuous action by a group $G$. There exists a unique topology on $X/G$ such that the quotient map $\pi :X\to X/G$ is continuous and also a local homeomorphism. Moreover, this map is open.

In this topology, a set $S\subseteq X/G$ is open if and only if its preimage in $X$ is open. If a subset $U\subseteq X$ is an open set that is disjoint from $U\cdot g$ for $g\ne 1$, then the restriction $\pi {|}_U:U\to \pi (U)$ is a homeomorphism.

The space $X/G$ with this topology is locally Hausdorff in general. We can assert that it is Hausdorff in the following case:

Lemma
Under the conditions of the previous proposition, the space $X/G$ is Hausdorff if and only if the image of the map

is closed in $X\times X$.
$◼$

We say that a group action is differentiable when the associated homeomorphisms are also diffeomorphisms.

Theorem (Quotient manifold theorem)
Suppose a Lie group $G$ acts smoothly, freely, and properly on a smooth manifold $M$. Then the orbit space $M/G$ is a topological manifold of dimension $dim(M)-dim(G)$, and has a unique smooth structure with the property that the quotient map $\pi :M\to M/G$ is a smooth submersion.
$◼$
See @lee2013smooth theorem 7.10 at page 153.

Lie group action

In case $G$ is a Lie group action on a manifold $M$ we have an injection

Indeed, $Diff(M)$ is in some sense like a (infinite dimensional) Lie group, whose Lie algebra is $\mathfrak{X}(M)$. This is due to the flow theorem for vector fields.

Given a vector in the Lie algebra $\mathfrak{g}$ we can consider the fundamental vector field on $M$. This corresponds to the differential of $\tau$ at the identity $e\in G$:

This is the induced Lie algebra action.