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.

The above gives that any finitely generated abelian group is the direct product of cyclic groups.

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