In my algebraic topology course we have been studying covering spaces. There are two group actions on any fiber: the left action of the group of deck transformations and the right action of the fundamental group. There are various propositions proved about general group actions that will later be applied. In particular,
Let $X$ be a transitive left $G$-set. If a subgroup $H$ of $G$ is the stabilizer of a point in $X$, then $X$ is $G$-isomorphic to the left-coset space of $H$, i.e., the collection of all left-cosets of $H$, where $G$ acts on a coset by left-translation.
Here is my question: How do I translate this into a statement about right actions? Is it really so simple as flipping every occurence of 'left' to 'right'?
Here is what I know: Given a right action of a group $G$ on a set $X$, we can define a left action as $gx := xg^{-1}.$ Also, any right action of $G$ on $X$ gives a left action of $G^{\text{op}}$ on $X$ and every group is isomorphic to its opposite via inversion.
The Wiki article on group actions makes the claim that "…only left actions can be considered without any loss of generality." (Towards the bottom of the 'Definition' section.) This link seems to hint that the justification for such a claim lies in the fact that a group $G$ and its opposite are 'naturally' isomorphic. The problem is that I only have a vague idea as to what that means as I have only a very basic working knowledge of categories from my algebraic topology book. Though the 'rules for translation' that the latter link provides are helpful, I still feel like I don't have a very good understanding of what's going on.
I realize what I'm looking for is a sort of theorem about theorems, which is strange. But can anybody offer some insight? If possible, can somebody elaborate on what this natural isomorphism has to do with anything? Basically, is the translation really so simple?
Best Answer
Given a left action $(g,x) \to g.x$ of $G$ on $X$, you can define a corresponding right action by defining $(x,g) \to g^{-1}.x$.