The Jordan-Hölder theorem

The Jordan-Hölder theorem states that any finite group can be broken down into a sequence of simple building blocks called simple groups. These sequences, known as composition series, may differ in the specific order of these blocks but will always have the same collection of blocks when considering their types. In essence, the theorem guarantees that, despite different ways to decompose a group, the fundamental components remain consistent, providing a unique "fingerprint" for the group’s structure. This helps mathematicians understand and classify groups in a structured, reliable way.