Left multiplication action - Imported Knowlpedia Content

Proposition (Left multiplication action). Let $G$ be a group. Define a map $G\times G\to G$ by

g\cdot x := gx.

Then this defines a group action of $G$ on the underlying set of $G$ .

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

Context. This action is the input for Cayley's theorem: it produces an injective homomorphism from $G$ into a symmetric group by viewing elements as permutations of $G$ .