# Why lower case “a” for “abelian group” and upper case “C” for “Cauchy sequence”?

This has been bugging me.

Why is the lower case letter "a" used to spell "abelian group" when upper case letters are used to spell the terms, "Gaussian Integral", "Cantor set" or "Cauchy sequence"?

Don't know where else to ask.

• Interesting question! – b00n heT May 26 '15 at 11:48
• I think it's a matter of convention for Abelian groups, see the preface of the book books.google.co.in/… using both uppercase and lowercase A. – baharampuri May 26 '15 at 11:51
• This would be more appropriate on english.stackexchange.com or linguistics.stackexchange.com In any case it's not a math question, and is actually only very tangentially related to math (proper nouns turning into adjectives happen everywhere, not just in math). – Najib Idrissi May 26 '15 at 12:04
• I would say that if you care this not for pure curiosity but for your own writing, then being consistent is the best policy. – Megadeth May 26 '15 at 12:06
• When I know a term is after a person's name, I capitalize it regardless of what others may do, so I capitalize "Abelian" and never has anyone told me to stop. My personal suspicion is that fewer people have heard of Abel than of Gauss, Cantor, or Cauchy and therefore it is more often assumed that "Abelian" is not derived from someone's name and therefore should not be capitalized. – Todd Wilcox May 26 '15 at 13:07

Some references still write here Abelian group, and not abelian group, e.g., see here. However, I admit that most texts write it with a lower case. Perhaps "abelian" it is a so common property, that it became a real adjective. Also, Grothendieck's anabelian geometry is written with a lower case.

The question has been discussed also at MO here. And the following nice saying can be found at MSE here:

You know you've made it as a mathematician when they start using your name in lowercase.

• In fact, I would say it's a duplicate of that MO question. – Kimball May 26 '15 at 12:18
• @Kimball Or, to stay within MSE, more or less a duplicate of math.stackexchange.com/questions/4526/…. What seems to be new is, that even the negation, i.e. anabelian is formed. I did not see ungaussian, or uneuclidean – Dietrich Burde May 26 '15 at 12:20
• @DietrichBurde Non-Euclidean is fairly common. – mrp May 26 '15 at 13:12
• @mrp Oh yes, with "non" fairly everthing is fairly common: non-abelian, non-Riemannian, non-Euclidean. But with "un"- or "an" ? – Dietrich Burde May 26 '15 at 13:13
• Who's Anabel? ;-) – Jeppe Stig Nielsen May 26 '15 at 21:17

This is an interesting question, in French the rule is clear when you use the proper noun you write with an upper case for instance "Cauchy sequence" is written "suite de Cauchy", "Cantor set" is written "ensemble de Cantor". However when you "adjectify" (It is certainly not the good word, sorry) a proper noun you just stop to put an upper case "abelian group" is written "groupe abélien" and "Gaussian integral" is written "intégrale gaussienne".

In English http://en.wikipedia.org/wiki/Capitalization the rules seems to be to always put an upper case when the adjective is derived from a proper noun.

So to answer your question, it looks like a gallicism to me.

• Of course the German rules are even stranger: Abel'sche Gruppe, Gauß'sche Glockenkurve, Ohm'sches Gesetz versus abelsche Gruppe, gaußsche Glockenkurve, ohmsches Gesetz – Hagen von Eitzen May 26 '15 at 12:08
• My language isn't great. So after reading that bit on the wikipedia article it seems writing "abelian" would be wrong to the pedantic. – Ishfaaq May 26 '15 at 12:09
• @Ishfaaq What is your native language then ? – Dietrich Burde May 26 '15 at 12:19
• @DietrichBurde Sinhalese, en.wikipedia.org/wiki/Sinhalese_language and Tamil en.wikipedia.org/wiki/Tamil_language. I speak both. Education is mostly in English though. . – Ishfaaq May 26 '15 at 12:21
• @Ishfaaq So how would you write "abelian group" in, say, Tamil ? – Dietrich Burde May 26 '15 at 12:23

This has been bugging me.

For bug-related problems, please use StackOverflow ;-$)$

Why lower case “a” for “abelian group” and upper case “C” for “Cauchy sequence” ?

Because Cauchy is a proper name, whereas abelian is an adjective. If it would have been called “Abel group”, then capitalization would have been mandatory. You can, of course, capitalize the adjective Abelian also, if you so desire, but, as I said, it's not mandatory.

• The first contentious example in the body of the question was "Gaussian" which is also an adjective and is almost always capitalised. – Ishfaaq May 26 '15 at 13:16
• @Ishfaaq: As I said, it is a matter of convention. You don't have to capitalize such adjectives, but you can, if you so wish. – Lucian May 26 '15 at 13:23
• @Ishfaaq: Also, notice that Gaussian integrals and abelian groups belong to very distinct mathematical fields, the former being used in calculus and statistics, and the latter in abstract algebra. Mathematicians, as a whole, do not exactly form some sort of monolithic group of people, which is why different standards and conventions arise in different areas of mathematics. – Lucian May 26 '15 at 13:34
• But I thought "Cauchy" in "Cauchy sequence" was also an adjective. – Emily May 26 '15 at 19:41
• @Arkamis: It is a noun $($or substantive$)$, acting as an attribute. The former is a part of speech, the latter is a grammatical modifier. – Lucian May 27 '15 at 3:26