I've encountered many different definitons of principal $G$ bundle from Morita's Geometry of differential forms , Hamilton's Mathematical gauge theory , Kobayashi and Nomizu's Foundations of differential geometry and wiki . (I'm crazy...)
But it seems that in different literature , they have such common properties (some are explaned as definitions, some are explaned as propositions, theorems or corollaries) .
Here is what I select : ( Suppose a fibre bundle is $(P,\pi,M,G)$)
Property 1 The action of $G$ on itself is left translation
Property 2 The action of $G$ on the total space $P$ to the right is free and smooth
Property 3 The quotient space have such identity : $P/G=M$
Property 4 The trivializing map satisfies $G-equavalent$ condition, whcih means :$\phi:\pi ^{-1}(U_\alpha)\cong U_\alpha\times G$ , then $$\phi(ug)=\phi(u)g $$ for $g\in G,u \in \pi ^{-1}(U_\alpha)$
Property 5 The action of $G$ preserves the fibres of $\pi$ and is simply transitive
It seems that if you select some of it, then it becomes a new definition of principal $G$-bundle. I'm not sure whether these properties are equivalent or not.So can anyone else show me the connections with each definitions? Thanks for your advice!