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!

$G$-torsor isomorphismbetween $X$ 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