[Verse 1] When every element plays nice and switches places clean Normal subgroups dance where conjugation's serene If gN equals Ng no matter what g you choose Then N is normal in G, that's the golden rule we use [Chorus] Normal means the cosets commute Quotient groups are the substitute Collapse the normal to identity See the structure's clarity G over N, what remains When you erase the smaller chains [Verse 2] In abelian groups every subgroup behaves But non-abelian worlds have subgroups that misbehave Conjugate and check if gNg-inverse stays the same If N transforms to itself, normal is its name [Chorus] Normal means the cosets commute Quotient groups are the substitute Collapse the normal to identity See the structure's clarity G over N, what remains When you erase the smaller chains [Bridge] Take gN times hN, multiply the cosets clean Gets you ghN precisely, operation's crystalline Well-defined because N is normal, that's the key Controlled blindness shows the structure you can see [Verse 3] Integers mod n show the pattern crystal clear Multiples of n vanish, remainders reappear Zero through n-minus-one, addition wraps around Quotient captures essence when the normal's been unwound [Chorus] Normal means the cosets commute Quotient groups are the substitute Collapse the normal to identity See the structure's clarity G over N, what remains When you erase the smaller chains [Outro] Choose what stops existing, let the normal disappear Large-scale structure emerges, details engineered to clear Quotient groups are telescopes for algebra's design Focus past the normal noise, see patterns redefined
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