[[Group theory MOC]]
# Abelian group

A [[group]] $A$ is said to be **abelian** if all its elements commute. #m/def/group

## Properties

- An abelian group has the same structure as a $\mathbb{Z}$-module. See [[Abelian groups as Z-modules]].
- An abelian group is equal to its [[Centre of a group|centre]], as well as the [[Centralizer in a group|centraliser]] of every element.
- [[Irreps of abelian groups are 1-dimensional]]
- [[Kernel of a homomorphism into an abelian group]]
- All subgroups of an abelian group are [[Normal subgroup|normal]]

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#state/tidy | #lang/en | #SemBr