Today I was browsing through Tao and Vu's additive combinatorics. They give the following definition:
Definition 3.1 (Torsion) If Z is an additive group and x ∈ Z, we let ord(x) be the least integer n ≥ 1 such that n · x = 0, or ord(x) = +∞ if no such integer exists. We say that Z is a torsion group if ord(x) is finite for all x ∈ Z, and we say that it is an r-torsion group for some r ≥ 1 if ord(x) divides r for all x ∈ Z. We say that Z is torsion-free if ord(x) = +∞ for all x ∈ ZIn the ensuing sections they often refer to "a torsion group". But they never use "torsion" alone, or if they do I didn't see it. This seems to me to be good evidence that the use of bare "torsion" isn't universal. (It also sounds weird to my ears, but a lot of things this particular professor said sounded weird to my ears and turned out to be standard usage among mathematicians. Since I was a first-year I had some learning to do.)
And the way that the word "torsion" is used here seems different than, say, an "open set". You can start a proof by saying "Let U be open." But "Let G be torsion." seems lazy. Perhaps mathematical English has two sorts of adjectives -- adjectives of the second kind like "open", which can be used without the implied noun they modify, and adjectives of the first kind like "torsion", which can only be used with the implied noun.
(The swapping of "second" and "first" is deliberate; it's like the Stirling numbers. I can never remember which is which.)