Structure Theorem for Finitely Generated Abelian Groups

The Structure Theorem for finitely generated abelian groups states that any such group can be broken down into a combination of simpler, well-understood building blocks: free parts (like copies of the integers) and finite parts (like cyclic groups of prime power order). Essentially, this theorem says every finitely generated abelian group can be expressed as a direct sum of these basic components, making its structure explicit and easier to analyze. It's a fundamental classification that provides a complete description of all possible structures such groups can have.