## 11 June 2008

I had an algebra professor my first year of grad school who would describe groups in which all elements had finite order as "torsion". For example, he'd say something like "since S4 is torsion"...

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 xZ, 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 xZ, and we say that it is an r-torsion group for some r ≥ 1 if ord(x) divides r for all xZ. We say that Z is torsion-free if ord(x) = +∞ for all xZ
In 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.)

thecooper said...

It doesn't seem weird at all to me. After all, when you say "let U be open", what you mean is "let U be an open set". I don't see the difference between that and the case of torsion.

That reminds me--I've read plenty of instances of the phrase "in the torsion case".

How do you feel about using torsion as a noun, as in "If there is torsion in H_2. . ."?

Isabel Lugo said...

"Torsion" as a noun seems more reasonable to me, because "torsion" is a noun in ordinary (non-mathematical) English.

complexzeta said...

Do people really talk about nonabelian groups being torsion? It's a bit awkward having ``torsion part'' (elements of finite order) not being a subgroup.

Isabel Lugo said...

I don't know about people; it might just be person.

intrinsicallyknotted said...

I'd say "torsion" by itself is a noun, and it does sound awkward using it as an adjective. There are plenty of instances in the English language where we use a noun as a modifier for another noun, but you wouldn't use the same noun as an adjective outside of the phrase.

On the other hand, in mathematics we have several cases where a person's name is used as an adjective; for example, a sequence can be Cauchy. Sounds awkward to me, but I also think it must be a pretty high honor to have one's name not only attached to a theorem or concept, but also used as an adjective. (The only thing better would be to have it uncapitalized, as in "abelian".)

D. said...

Put my vote in the "not awkward to say 'Let G be torsion'" camp. To me it stems quite naturally out of talking about G-torsion (meaning the part of G that has finite order).

d

Anonymous said...

what grates *my* ear most is
can a sequence be cauchy?

for me, yes ... *informally*.
i'd prefer to avoid *writing*
"let G be torsion" *or* (especially)
"let S be cauchy".

(for that matter, can a threat be death?
can a note be suicide?)

i first became aware of this
kind of thing, if memory serves,
was starting to catch on (still
known as "conservation" in those days)
and some teacher told us about
"air pollution" and "water pollution"
... and "noise pollution".
well, doggone it (sez i),
it's the *air* and the *water*
that get polluted on the one hand,
but surely it can't be the *noise*
that's getting polluted on the other.

the thing is, it *sounded really stupid*.
so i was doggoned it i'd say it.
somebody (joe kenny; i haven't seen
him since that year, alas) soon
pointed out that on my model,
one would allow "blood poisoning"
(consistency and usage don't mix.)

Anonymous said...

I'm of the philosophy that if you're going to write something other people are going to read, you should just do it right. Thus "Let U be an open subset of X" and "Let G be a torsion abelian group". I mean, are keystrokes so valuable that you want to save them? (And I would only use the word torsion for abelian groups.)

Anonymous said...

I've definitely heard folks use "torsion" as an adjective before. I think it is probably a reasonably common usage. I was just reading a paper just posted on the preprint arxiv that uses the term in this way:

http://arxiv.org/abs/0806.2068

Emmanuel Kowalski said...

It would be helpful to know if "torsion" came originally into English mathematical usage from another language. For instance, in French, one definitely says very naturally "soit G un groupe de torsion", or one can conclude an argument with "et donc G est de torsion".

A French writer, knowing how this type of construction "un X de Y" often translates into "a Y X" into English, would then assume that the most natural translations in English are "let G be a torsion group", and "and hence G is torsion"...

The exact same thing happens with Cauchy sequences: in French, one speaks of "une suite de Cauchy", and can write "et donc (u_n) est de Cauchy", which it seems natural to translate as "a Cauchy sequence" and "and hence (u_n) is Cauchy". Whether this is good English is of course another question...

(Incidentally, about Cauchy, in analytic number theory, "to Cauchy" is a verb, meaning "to apply Cauchy's inequality" to some well-chosen sum...)

Emmanuel Kowalski said...

P.S. I forgot to mention that, maybe not coincidentally, the preprint mentioned in the comment before mine is by a French mathematician.

Anonymous said...

My professors at Berkeley also used the "Let X be torsion" construction.

Isabel Lugo said...

Emmanuel,

that's a good point, that what sounds a bit off to me as a native speaker of English might be reflecting the workings of some other language.

The person I associate with the use of "torsion" as an adjective is a Romanian who was trained in Germany.