Abelian group classification

An Abelian group classification involves categorizing groups where the order of combining elements doesn't matter—called Abelian groups. These groups can be broken down into building blocks: cyclic groups (generated by a single element) and their direct sums. Mathematicians have shown that every finitely generated Abelian group can be expressed as a combination of simple, well-understood components like integers and finite cyclic groups. This classification helps in understanding the structure of these groups systematically, similar to how prime factorization breaks numbers into fundamental parts.