I am having difficulty with the following problem.
Show that, a finite non-abelian simple group cannot have an abelian subgroup of prime power index.
What I was thinking is that I can somehow use a result by Burnside which says that "For a finite group G having a conjugacy class of order prime power then G cannot be simple"
Kindly suggest how to proceed further.
Best Answer
Take a non-identity element $a$ in $H$, then the centralizer of $a$ contains $H$ since $H$ is abelian. Then the index of the centralizer of $a$ in $G$ is of prime power, So by Burnside theorem (you stated), centralizer of $a$ is the whole group, which shows that the centre of $G$ is nontrivial. Simplicity implies $G$ has to be abelian, which is contraction to the given hypothesis. Hence a non-abelian simple group cannot have an abelian subgroup of prime power index.