अबेलिया इसके अंग्रेजी अर्थ का उदाहरण

This is an abelian group for all k.

In this case, X has the structure of a commutative algebraic group (with p0 as the zero element), and so the set X(k) of k-rational points is an abelian group.

The Mordell–Weil theorem says that for an elliptic curve (or, more generally, an abelian variety) X over a number field k, the abelian group X(k) is finitely generated.

Its termination in general would follow from the conjectures that the Tate–Shafarevich group of an abelian variety over a number field is finite and that the Brauer–Manin obstruction is the only obstruction to the Hasse principle, in the case of curves.

The strongest known result on the Bombieri–Lang conjecture is Faltings's theorem on subvarieties of abelian varieties (generalizing the case of curves).

Namely, if X is a subvariety of an abelian variety A over a number field k, then all k-rational points of X are contained in a finite union of translates of abelian subvarieties contained in X.

(So if X contains no translated abelian subvarieties of positive dimension, then X(k) is finite.

Any Weil cohomology theory factors uniquely through the category of Chow motives, but the category of Chow motives itself is not a Weil cohomology theory, since it is not an abelian category.

such that \mathfrak{g}_{i+1} is an ideal in \mathfrak{g}_i and \mathfrak{g}_i/\mathfrak{g}_{i+1} is abelian.

A solvable nonzero Lie algebra has a nonzero abelian ideal, the last nonzero term in the derived series.

Every abelian Lie algebra \mathfrak{a} is solvable by definition, since its commutator [\mathfrak{a},\mathfrak{a}] 0.

There is a generalisation of Serge Lang to abelian varieties (Lang, Abelian Varieties).

This is a generalization of the result of Frobenius that commuting matrices are simultaneously upper triangularizable, as commuting matrices form an abelian Lie algebra, which is a fortiori solvable.