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