[Verse 1] When functions map from group to group They might preserve what matters most Phi takes a-b and keeps the loop ab maps to phi-a times phi-b host [Chorus] Homomorphism keeps operation alive Structure flowing from left to right Kernel catches what won't survive Image shows what makes the flight When it's bijective, isomorphic light Same group wearing different disguise [Verse 2] Identity maps to identity Inverse maps to inverse true Kernel holds the mystery Elements that vanish through Normal subgroup living there While image builds a faithful crew [Chorus] Homomorphism keeps operation alive Structure flowing from left to right Kernel catches what won't survive Image shows what makes the flight When it's bijective, isomorphic light Same group wearing different disguise [Bridge] First theorem speaks the deeper truth G mod kernel equals image proof Quotient by what's identified Gives exactly what's inside Second third theorems extend the dance HN over N gets its chance H mod intersection stands Third shows quotient of quotients lands [Verse 3] When injection fails some merge together Kernel measures redundancy Quotient cuts the binding tether Image shows what's meant to be Collapsing sameness sets us free Universal pattern, can't you see [Final Chorus] Homomorphism keeps operation alive Structure flowing from left to right Kernel catches what won't survive Image shows what makes the flight Quotient kernel equals image sight Algebra's most profound design [Outro] Same group, different names Isomorphic, plays the same games
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