[Verse 1] Every abelian group that's finitely made Has a secret structure hiding in the shade Break it down to cyclic pieces, neat and clean Z to the r times torsion, what does that mean? Free part floating, torsion bound tight Invariant factors climbing left to right [Chorus] Classification complete, no mystery remains Every group's a product of cyclic chains Z mod n divides the next in line Elementary divisors, powers of prime Simple groups are atoms, can't be split Building blocks of everything that fits [Verse 2] Simple means no normal subgroups inside Just the trivial ones with nowhere to hide Jordan-Hölder says they're fundamental bricks Every finite group's built from these tricks Cyclic primes and alternating dance Sixteen Lie families in their stance [Chorus] Classification complete, no mystery remains Every group's a product of cyclic chains Z mod n divides the next in line Elementary divisors, powers of prime Simple groups are atoms, can't be split Building blocks of everything that fits [Bridge] Twenty-six sporadic outliers stand alone Monster group with order astronomically grown Eight times ten to fifty-three so vast Fifteen thousand pages, the theorem's cast Two thousand four, the proof complete Mathematics' greatest classification feat [Outro] From abelian structure to simple core Group theory's secrets hidden no more Torsion and free, divisors align The universe of groups falls into line
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