Commas between adjectives

Question

Is it correct to use these expressions without commas (as written below)?

1. A finite simple group.
2. A finitely generated abelian group.
3. A finitely generated torsion-free abelian group.
4. A finitely generated periodic infinite group.

From my experience, I can't remember ever seeing commas in any expressions of this kind (in mathematical texts). But that could be because I simply wasn't paying attention to punctuation.

I've spent some time googling for the appropriate rule, and the rule says that commas should be avoided between cumulative adjectives, and should be used between coordinate adjectives. To distinguish between the two, the following test is usually suggested: if the adjectives can be rearranged, and also the phrase doesn't look weird when "and" is placed between them, then these adjectives are coordinate.

When I try to apply the rules to cases 1-3, I feel a certain amount of doubt. In case 1 I could agree that "a simple finite group" sounds unusual. In case 2 I could also agree that "abelian group" is a very common expression, so it feels natural to place "abelian" after all other adjectives.

But what about case 4? The three modifiers seem to be more or less on equal footing. It doesn't look weird at all when they are rearranged or when "and" is tossed in: "an infinite, periodic, and finitely generated group" - looks fine to me.

So, my questions are:

1. Are examples 1-4 above correct or not? Why?
2. Is there a convention to avoid commas between adjectives in mathematical texts altogether (ignoring the general English rule)?

Thanks.

Answer 1

1) In my opinion they look fine. Part of the issue is that only a few disciplines in writing ever have the opportunity to link together so many adjectives, and mathematics is one of them.

I could cook up some excuses why using no commas would be appropriate, but ultimately I think the better argument is that if we rigidly enforced rules that made use use lots of commas, there would simply be too many commas. In general, overuse of commas disrupts cognitive flow while reading.

2) There probably is, but I imagine it's unwritten. I've never heard or read advice about this rule. You described that in your experience you don't recall commas being used this way, and I would say the same thing. Finally, it is unlikely that the absence of commas in this particular location will draw attention. There are far more egregious writing problems that will do that.