A principal $G$-bundle, where $G$ denotes any topological group, is a fiber bundle $\pi :P → X$ together with a continuous right action $P × G → P$ such that $G$ preserves the fibers of $P$ and acts freely and transitively on them.

A principal $G$-bundle, where $G$ denotes any topological group, is a fiber bundle $\pi :P → X$ together with a continuous right action $P × G → P$ such that $G$ preserves the fibers of $P$ and acts freely and transitively on them.

An equivalent definition of a principal $G$-bundle is as a $G$-bundle $\pi :P → X$ with fiber $G$ where the structure group acts on the fiber by left multiplication. Since right multiplication by $G$ on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by $G$ on $P$. The fibers of $\pi$ then become right $G$-torsors for this action.

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