Abelian group

In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that

Examples

Every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. In particular, the integers Z form an abelian group under addition, as do the integers modulo n Z/nZ.

Related Topics:
Cyclic group - Integer - Integers modulo ''n''

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The real numbers form an abelian group under addition, as do the non-zero real numbers under multiplication.

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Every ring is an abelian group with respect to its addition operator. Also, in every commutative ring the invertible elements, or units form an abelian multiplicative group.

Related Topics:
Ring - Commutative ring - Units - Multiplicative group

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Any subgroup of an abelian group is normal, and hence factor groups can be formed at will. Subgroups, factor groups, products and direct sums of abelian groups are again abelian.

Related Topics:
Subgroup - Normal - Factor group - Products - Direct sums

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