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. 
 A: 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.
A: 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.

A: 
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.
