# a question on the correct terminology for $G$-torsors

Fix a group $G$. Recall a space $X$ is a $G$-torsor if $G$ acts freely and transitively on $X$ (if $G$ is non-abelian one has to talk about left or right torsors).

A choice of a point $x \in X$ induces a bijection between $X$ and $G$, and hence $X$ is non-canonically isomorphic to $G$.

Is there a universally accepted name for such a choice, as in:

"Let $X$ be a $G$-torsor, and choose a [insert name here] between $X$ and $G$..."

The word that comes naturally to mind is affine map, but this somehow isn't appropriate when $G$ has torsion, say...

I note that the wikipedia entry:

http://en.wikipedia.org/wiki/Principal_homogeneous_space

does not give a name to such a map; neither does the lovely little expository article by Baer:

http://math.ucr.edu/home/baez/torsors.html

Apologies if this question isn't deemed appropriate for mathoverflow - it's hardly a research level question I know, but it's not the sort of thing I could really ask anywhere else!

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With Dan Petersen, I would probably say "choose a trivialization of $X$". But if you want to fit it into your sentence, "Let $X$ be a $G$-torsor, and choose a $G$-torsor isomorphism between $X$ and $G$" works fine. Note also that in other categories, and notably in the category of bundles over $Y$, one can easily talk about torsors, but it is not true that every torsor has a global element = trivialization. Anyway, "global element" or "global section" are fine words to adopt. – Theo Johnson-Freyd Apr 5 '11 at 19:31

I would call it a trivialization of the torsor. This terminology comes from the relative situation when one deals with principal bundles rather than principal homogeneous spaces. In this case, a trivialization would be a global section of the principal bundle $X \to Y$ (rather than a point of X), so a trivialization is the same thing as an isomorphism between X and the trivial bundle $G \times Y \to Y$.