Group Action

left action

A (left) action of Group $G$ on set $X$ consists of a set map:

σ : G \times X \to X

Where $(g, x) \mapsto σ (g, x) = g \cdot x = g x$ such that:

$g_{1} \cdot (g_{2} \cdot x) = (g_{1} \cdot g_{2}) x$ for all $g_{1}, g_{2} \in G, x \in X$
$1_{G} \cdot x = x$ for all $x \in X$

This definition may seem a bit loose, so mathematicians sometimes use another definition:

left action

A (left) action of $G$ on $X$ is a group Homomorphism (or Group Morphism) (map) $σ : G \to S_{X} = {f X \to X | f is a bijection}$ (see Symmetric Group & Permutations)

We'll later see that these definitions are equivalent. However, let's first look at an example.

Often we'll use $σ_{g}$ to represent how $σ (g) = σ_{g}$ in the alternative definition above. Namely:

σ_{g} (x) = g \cdot x

An Example

The trivial action of $G$ on $X$ is $σ : G \to S_{X}$ such that each $g \mapsto 1$ . Namely $g x = x$ for all $g \in G, x \in X$ .
The Dihedral Groups, like $D_{8}$ for the symmetries of a square, acts on the set of vertices of that square ( $X = {v_{1}, v_{2}, v_{3}, v_{4}}$ for simplicity, for example $r v_{1} = v_{2}$ ). Define $σ_{r} = (v_{1} v_{2} v_{3} v_{4}) \equiv (1 2 3 4)$ . Then this is a group action, where $σ : D_{8} \to S_{X} ≅ S_{4}$ where $σ_{r} : r \mapsto (1 2 3 4)$ .

$D_{8}$ can also act on the set $Y = {d_{1}, d_{2}}$ , the diagonals of the square:

Here $σ_{r} = (d_{1} d_{2}) \equiv (1 2)$ . Here $D_{8} \to S_{Y} ≅ S_{2}$ .

One action of the group onto itself is left multiplication by the group operation:

g ⋆ g^{'} = g g^{'}

Note

Why left multiplication? Namely, it's because our definition is in terms of terms from the group being on the left.

Is there any group elements that keeps each element fixed? Namely is there some $g \in G$ such that $g ⋆ x = 1_{G}$ for all $x \in X$ ? See the answer on Trivial Group Action.