After Cauchy's theorem, the next natural question is how far existence can be pushed when the group is abelian. The focus keyword is subgroups of finite abelian groups, because abelian groups allow stronger subgroup existence results than general finite groups. In a finite abelian group, commutativity lets cyclic pieces combine without the obstructions that occur in nonabelian settings. This lesson proves the converse of Lagrange's theorem for finite abelian groups and prepares the prime-power language used in p-groups and Sylow theory.

Let $G$ be a finite group. Then $G$ is called a \textbf{finite abelian group} if $G$ has finitely many elements and $$\begin{equation*} ab=ba \end{equation*}$$ for all $a,b\in G$.