# why do we write abelian group instead of Abelian group?

Suppose an object (or a concept or ...) is named after the person X, in honor of Mr. or Mrs. X in mathematics: X-ian objects/ X-ic objects/ X objects.

It is natural for me to write the first letter of his/ her name in English in capital letters when referring to that object/ concept. For instance: Gaussian curvature, Newtonian mechanics, Riemannian geometry, Archimedean valuation, Eulerian graph, Platonic solids, Jacobian matrix, Noetherian rings, Artinian rings, Hamiltonian path, Hermitian matrix, Hessian matrix, ...

But why do we usually write abelian group instead of Abelian group? Or why do we usually write abelian variety instead of Abelian variety?

My strong suspicion is that perhaps the importance of abelian groups has reached us through mathematicians in languages other than English. But since I do not have a background in mathematical history, it is very likely that my guess is wrong.

Is this related to the French school of mathematics? (I know that "variety" is the French equivalent of "Manifold".) If yes, then why do we write Galois extensions?

Are there any exceptions other than abelian groups?

Also, I do not know what is the suitable tag for my question.

• I don't know the answer to this, but interestingly enough this morning I was writing a bibliographic entry in one of my manuscripts I tinker with, and the paper's title included "abelian group". However, the photocopy of the paper I had wrote the title in all capital letters (e.g. $\ldots$ ABELIAN GROUP $\ldots),$ and my policy is to italicize titles, capitalize the first word, then only capitalize any later words if they would nearly always be capitalized. I wasn't sure whether to capitalize abelian, so I looked at some bibliographies in my books and searched online before deciding on 'a'. – Dave L Renfro Dec 27 '20 at 18:00