They are used for studying finite flat commutative group schemes.

The equivalence class forms a commutative group under operation .

It forms the commutative group with operation defined as binary vector addition.

They all carry a commutative group law.

Commutative group theory in 11 words.

All characters of a finite commutative group.

It is well known that the unipotent commutative groups correspond to Dieudonné modules.

Common cases include fppf sheaves of commutative groups over S, and complexes thereof.

For example, commutative groups and (arbitrary) groups are two different species of the same echelon construction scheme.

More compactly, an abelian group is a commutative group.

Pontryagin duality provides a large supply of examples of compact commutative groups.

In more contemporary language this appears as part of the theory of algebraic groups, dealing with commutative groups.

The Pontryagin duality describes the theory for commutative groups, as a generalised Fourier transform.

Restriction of scalars over an inseparable field extension will yield a commutative group scheme that is not a torus.

The difference between rings and semirings, then, is that addition yields only a commutative monoid, not necessarily a commutative group.

Néron models exist as well for certain commutative groups other than abelian varieties such as tori, but these are only locally of finite type.

A semiabelian variety is a commutative group variety which is an extension of an abelian variety by a torus.

An abelian group, or commutative group is a group whose group operation is commutative.

In other words, the maps form a commutative monoid (in the cases and ) or a commutative group (in the cases and ).

As a torus, J carries a commutative group structure, and the image of C generates J as a group.

Classification of simple Lie groups One can show that the fundamental group of any Lie group is a discrete commutative group.

Finite flat commutative group schemes over a perfect field k of positive characteristic p can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting.

If G is a constant commutative group scheme, then its Cartier dual is the diagonalizable group D(G), and vice versa.

In Pontryagin duality theory for locally compact commutative groups, the dual object to a group G is its character group which consists of its one-dimensional unitary representations.

An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity).