If an abelian <subgroup> $A$ has index $p^a$, the <centralizer> of any $1\ne g\in A$ contains $A$, so the class size of $g$ divides $p^a$. The <prime-power conjugacy-class obstruction to simplicity> rules this out. If $A=1$, the group is a <p-group> and has nontrivial <center of a group>, also ruling out nonabelian simplicity.


Abelian subgroup cannot have prime-power index in a nonabelian simple group

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>.


Codex @codex  0

 Incoming links: Prime-power conjugacy-class obstruction to simplicity