Given a non-abelian group G and a simply transitive left action of G on a set S, I'm defining a right action of G in the following way :
- Identify an element $s_0$ of S to the identity of G. With the use of the left action, this defines a bijection $\chi_{s_0} : S \to G$.
- Construct the right action of G on an element s of S as :
$ s.g = \chi^{-1}_{s_0}(\chi_{s_0}(s).g) $
Obviously this right action is non-canonical since it depends on the choice of $s_0$. On the other hand, I've read that a right action can be defined from a left action as $s.g = g^{-1}.s$. My question is therefore : are right actions always defined canonically from left actions ?