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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.

4 votes

Principal bundles over groups

Colliot-Thélène's paper Résolutions flasques des groupes linéaires connexes, J. für die reine und angewandte Mathematik (Crelle) 618 (2008) 77--133, http://www.math.u-psud.fr/~colliot/resolflsq_211107 …
Mikhail Borovoi's user avatar
6 votes
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What are the symmetries of a principal homogeneous bundle?

No, in general $G=G(\mathbf{Q})$ can be strictly smaller that ${\rm Aut}(\mathbf{Q})$. Let $G$ be a Lie group and $H\subset G$ be a Lie subgroup. Set $P=G$, $\ Q=G/H$, and define the maps in the obv …
Mikhail Borovoi's user avatar