---- > [!definition]+ Definition. ([[abelian group]]) > An **abelian group** is a [[group]] whose [[binary operation]] is commutative. > > Abelian groups are objects of the [[category]] $\mathsf{Ab}$, the full [[subcategory]] of $\mathsf{Grp}$ obtained by keeping just the objects which are abelian (the homsets are untouched). ^definition > [!note] Remark. > When working with abelian groups, often one trades the conventional 'multiplicative notation' $ab, a^{n}, 1, \text{ etc.}$ for 'additive' notation $a+b, na, 0, \text{etc.}$ Especially if some 'multiplicative' structure simultaneously exists on the set (like for rings). ^note > [!generalization] > [[Every abelian group is a Z-module, in exactly one way]]. Thus, "[[abelian group]]" and "$\mathbb{Z}$-[[module]]" are one and the same notion. ^generalization > [!basicproperties] The Category $\mathsf{Ab}$. > - Like in $\mathsf{Grp}$, trivial groups $(e)$ are [[terminal object|zero objects]] in $\mathsf{Ab}$. > - Unlike in $\mathsf{Grp}$, [[finite direct products and coproducts align in the category of abelian groups]], in the sense that $G \times H$ satisfies the [[universal property]] for [[categorical coproduct|coproducts]] in $\mathsf{Ab}$. ^properties ---- #### > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```