Left multiplication action

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

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.

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 $G$.