# Statement(s) > [!NOTE] Statement 1 (Fundamental Theorem of Finite Abelian Groups) > Let $G$ be a [[Finite Group|finite group]] [[Groups|abelian]] group with order gt;1.$ Then there are integers $b_{1},b_{2},\dots,b_{n}>1$ such that $b_{1}\mid b_{2};\, b_{2} \mid b_{3};\; \dots; b_{n-1}\mid b_{n}$ and $G\cong C_{b_{1}}\times C_{b_{2}}\times \dots \times C_{b_{n}}$where $\cong$ denotes [[Homomorphisms of groups|isomorphism]] and $G\cong C_{b_{1}}\times C_{b_{2}}\times \dots \times C_{b_{n}}$ denotes a [[Cartesian Product|cartesian product]]. **Remark**: It follows from [[Product Rule for Counting (Fundamental Counting Principle)]] that $\#G = b_{1}b_{2}\cdots b_{n}.$ # Proof(s) **Proof of statement 1:** ... $\blacksquare$ # Application(s) **Consequences**: **Examples**: # Bibliography