= Finite abelian group
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A <finite abelian group> combines a finite <set> of reversible operations with <commutativity>. All <finite cyclic groups> qualify, but <direct products of groups> supply examples that need more than one <generator of a group>. The <Fundamental theorem of finitely generated abelian groups> says that every <finite abelian group> is a <direct product of groups> of finite <cyclic groups>; the finite case has no infinite cyclic factors.<Group Theory (v4.01) by James S. Milne>[Theorem 1.54]{ref}
These factors distinguish different ways to have the same total number of states. A single four-position dial gives $C_4$, while two independent binary switches give $C_2\times C_2$. Both have four states and <commutative> composition, yet only the dial permits a single operation whose repetition visits all four states. The following <Cayley tables> make that distinction visible without relying on the labels alone.
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