= Prime-power conjugacy-class obstruction to simplicity
A nonabelian <simple group> has no nonidentity <conjugacy class> of prime-power size. Coprime-degree <irreducible characters> vanish on such an element: the <conjugacy-class sum> makes the character-to-degree ratio an <algebraic integer>, and the <Kronecker theorem on algebraic integers in the unit disk> makes a nonzero ratio a <root of unity>, forcing a scalar in a faithful <group representation>. Column <character orthogonality> then makes $1/p$ an <algebraic integer>, impossible. This is the character-theoretic ingredient in <Burnside's theorem>.
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