# Abelian group

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## 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 ?
Every finite Abelian group is determined by its endomorphism monoid in the class of all groups.
Then we exploit the fact that for a nondegenerate pairing on an abelian group holds [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and that any r [member of] S acts on v by the 1-dimensional character [[chi].sub.j](r) = (r,[s.sub.j]):
Let H be a finite abelian group written additively and End(H) be the endomorphism ring of H.
Singh, "Some decomposition theorems on abelian groups and their generalisations.
Corollary 1.2 If G is an abelian group of order n and n is divisible by m then there exists a subgroup of G with order m.
In fact the two objects are not truly equivalent, in that the information contained in matroids over Z is richer, because there are many finite abelian groups with the same cardinality.
A subgroup H of an abelian group G is pure in G if nH = H (1 nG, where n is any non-zero integer.
Rankin, On the number of abelian groups of a given order, Quart J.
Throughout this brief article, let G be an abelian group with p-component of torsion [G.sub.p] and R a commutative ring with 1 (called also unital) of prime characteristic p with nil-radical N(R).
These very different groups R, [T.sup.1], Z are all abelian and locally compact ([T.sup.1] is compact), and so are united by working with locally compact abelian groups.
This note shows how to obtain an abelian group with an addition like operation (Joyner 2002:70-72) beginning with a subtraction binary operation.

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