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** [[Principal homogeneous space]] versus [[abelian group]];

{{r|Abelian group}}

{{r|Free Abelian group}}

{{r|Abelian group}}

{{r|abelian group}}

*The rational numbers form what mathematicians call an [[abelian group]] under addition. *When we exclude the number 0, they form an [[abelian group]] under multiplication as well.

...s of an abelian group are normal. As a matter of style, the operator on an abelian group is often called "addition" and the identity element called 0. Conversely, n By similar reasoning, the [[real number]]s with addition is an abelian group, but it is not cyclic. There is no single element such that every real numb

...roups were mainly studied in terms of [[permutation]]s. Some aspects of [[abelian group]] theory were also known in the theory of [[quadratic form]]s. ...' such that ''a'' * ''b'' ≠ ''b'' * ''a''. A group ''G'' is said to be ''[[abelian group|abelian]]'' (after the mathematician [[Niels Abel]]) (or ''commutative'') i

...'-units: the group of ''S''-units is finitely generated, with [[Rank of an abelian group|rank]] (maximal number of multiplicatively independent elements) equal to '

The fractional ideals of ''K'' form an abelian group under ideal multiplication with the fractional ideal ''O''_''K'' ''U'' is a finitely generated [[abelian group]] with [[free rank]] ''r''+''s''-1 and torsion subgroup the [[root of unity

.... Then ''E''(''K''), the points of ''E'' with coordinates in ''K'', is an abelian group. The structure of this group is determined by the Mordell-Weil theorem, wh

...[[whole number]]s can be assembled. Let us make this more precise. The [[abelian group]] <math>\mathbb{Q}^{+}</math> of positive rational numbers under multiplica

...a'' * ''b'' is identical to ''b'' * ''a'' – then the group is said to be [[Abelian group|Abelian]]. | Abelian group

...operties listed above for addition say that '''Z''' under addition is an [[abelian group]]. As a group under addition, '''Z''' is a [[cyclic group]], since every no

...>2h</sub> symmetry; D_2h is an [[Abelian group]] that has—as any Abelian group—only one-dimensional irreducible representations. Hence all electronic st

* (''R'', +) is an [[abelian group]]

...easily for finite groups.)</ref> ''Gal(L/K)'' of ''L'' over ''K'' is an [[abelian group]] -- are relatively well understood.

Properties 1 - 5 state that a vector space is an [[Abelian group]] with addition as group operation.

...ite just "''ax''". The first argument belongs to a field, the second to an abelian group ("ab_group_add") written additively (thus, "''x''+''y''" is the sum of ''x'