# Abelian group

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Related to Abelian group: group theory, vector space, Cyclic group

## Abelian group

[ə′bēl·yən ′grüp]
(mathematics)
A group whose binary operation is commutative; that is, ab = ba for each a and b in the group. Also known as commutative group.
References in periodicals archive ?
because each finite Abelian group is determined by its endomorphism semigroup in the class of all groups ([7], Theorem 4.
3 A(n, H) and B(n, H) are the subgroups of an additive Abelian group G.
Like a free abelian group a free group consists of words, but these behave as follows: (a) when deciding when two words are equal, we are not allowed to rearrange their letters and antiletters and we may only create or annihilate letter/antiletter pairs if they occupy adjacent positions (for example, ACC'B = ABD'D but ACBC' is not equal to CAC'B); (b) the product of two words is again obtained by juxtaposing them; (c) to obtain the inverse of a word we must not only replace each letter by the corresponding antiletter and vice versa but reverse their order (for example, the inverse of AB' is BA', not A 'B).
If Q is a medial quasigroup, then there exist an Abelian group <Q, +, -, 0>, its commuting automorphisms [phi] and [psi], and an element c [member of] Q such that
From [2] and [5], (M, *) forms an Abelian group with identity element [[mu].
v](y) for all y [member of] Y} and (FL(V,W), +) is an abelian group.
Let G be an Abelian group and let E be a Banach space.
d], and is also isomorphic to the free Abelian group on d generators.
Let G be an abelian group of order G and let H be a subgroup of order h.
e], for e [member of] ob(G), an abelian group, then A is maximal commutative in A [[?
Harinath, On the Cayley graph of an abelian group, Proceedings of National Workshop on Graph Theory and its Applications, Manonmaniam Sundarnar University, Tirunelveli, February 21-27, 1996.

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