Abelian group

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

Finite abelian groups

The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order. This is a special application of the fundamental theorem of finitely generated abelian groups in the case when G has torsion-free rank equal to 0.

Related Topics:
Prime - Fundamental theorem of finitely generated abelian groups

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

For example, Z/15Z = Z/15 can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: Z/15 = {0, 5, 10} ⊕ {0, 3, 6, 9, 12}. The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are isomorphic.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

For another example, every group of order 8 is isomorphic to either Z/8 (the integers 0 to 7 under addition modulo 8), Z/4  ⊕ Z/2 (the odd integers 1 to 15 under multiplication modulo 16), or Z/2  ⊕  Z/2  ⊕  Z/2.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~