Left multiplication action

Proposition (Left multiplication action). Let $G$ be a group. The formula

g\cdot x := gx.

defines a group action of $G$ on its underlying set.

Moreover, this action is:

transitive (there is one orbit), and
free (only the identity fixes any element),

hence it is a regular action, often called the left regular action.

Remarks

This action is the input for Cayley's theorem: it sends each element of $G$ to a permutation of $G$ , producing an injective homomorphism into a symmetric group.