Abelian subgroup of prime power index

Question

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.

Monotonic · 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.

Abelian subgroup of prime power index

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