Definition 1: A principal $G$-bundle is a fiber bundle $F \to P \xrightarrow{\pi} X$ together with a right action of $G$ on $P$ such that:
(i) $G$ acts freely and transitively on fibers.
(ii) $G$ preserves fibers.
I am trying to show that the structure group of a principal $G$-Bundle is $G$, following wikipedia's definition of a structure group of a fiber bundle. What I have come up with is the following:
Suppose $(U_i,φ_i)$,$(U_j,φ_j)$ are local trivilizations, then since $φ_i \vert_{π^{-1}(q)} : π^{-1}(q) \to \{q\}\times F $ (similarly for $j$) we should have for $x \in U_i \cap U_j $, $ξ \in F$ $$φ_i φ^{-1}_j (x,ξ)=(x,ξ') $$
Now $(x,ξ')$ corresponds to an element $p \in π^{-1} (x)$ by $φ^{-1}_j$ and $(x,ξ)$ to an element $p' \in π^{-1} (x)$. But we have a transitive right action of G on P, so that we can find a $g \in G$ s.t. $p'=pg$. Thus, for each $x \in U_i \cap U_j $ we can define a left action on $F$ that takes $ξ$ το $ξ'$ : $$φ_i φ^{-1}_j (x,ξ)=(x,ξ')=(x', t_{ij}(x)ξ) $$ and now $t_{ij}(x)=g \in G$ (the $g$ which acting on $p$ gives $p'$) acts on $F$, so that $G$ is the structure group of our principal $G$ bundle. Is this reasoning correct/complete? Have I missed something? I haven't managed to find anything in the literature proving that given Definition 1 we can show that the structure group is $G$ and I was wondering if what I came up with is the way it's supposed to be done.
EDIT Seeing this question Equivalence of Definitions of Principal G -bundle it seems to me from Defition 3, since existence of G-equivariant cover is equivalent to having a structure group G, that the structure group being G is part of the definition and thus does not follow from it? I have to say I am generally confused by the definition(s), does it contain both actions on P and on the fiber, or do we define one action and the other follows?
