a [[group]] is called **abelian** if the group operation is **commutative**, i.e. $\forall u, v \in X : u \cdot v = v \cdot u$ formalizes intuitions about addition ![[group]] Any subgroup of an abelian group is a [[normal subgroup]]: $ g_{\natural} h = g h g^{-1} = h \in H $ # fundamental theorem of finite abelian groups Every finite abelian group can be expressed as the [[direct sum of linear subspaces]] of cyclic subgroups of prime power order