Abelian group

Definition

An abelian group $G$ is a group whose operation is commutative.

Sometimes when working with abelian groups (almost exclusively in ring/module theory), we use additive notation instead of the usual multiplicative notation:

Thing	Multiplicative notation	Additive notation
Operation	$* . \times \circ$ or omitted	$+$
Identity	$1$	$0$
Inverse	$g^{-1}$	$--inline -g$
Group operation repeated $n \in \mathbb{Z}$ times	$g^{n}$	$ng$ (particularly in a ring)

Properties

TODO - simple group decomposition

Any abelian group can be considered as a module over $\mathbb{Z}$ . As a module, any finitely generated abelian group can be expressed as the direct sum of cyclic groups.

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all these pages adapted with probably insufficient credit from my university's lecture notes