Reduction of restricted Burnside problem to associated Lie ring

Statement
The following are equivalent for any natural number $$n$$ and any prime power $$p^r$$:


 * 1) The restricted Burnside group $$RB(n,p^r)$$ is a finite group
 * 2) The associated Lie ring $$L(B(n,p^r))$$ of the Burnside group $$B(n,p^r)$$ is a nilpotent Lie ring

Facts used

 * 1) uses::Cauchy's theorem which states that for any prime dividing the order of a finite group, the group has an element whose order is that prime.
 * 2) uses::Prime power order implies nilpotent
 * 3) uses::Equivalence of definitions of periodic nilpotent group: The key thing we use is that any finitely generated periodic nilpotent group is finite.
 * 4) uses::Nilpotency class of associated Lie ring equals nilpotency class of quotient of group by nilpotent residual

Proof
Given: Natural number $$n$$, prime number $$p$$, natural number $$r$$

To prove: $$RB(n,p^r)$$ is finite if and only if $$L(B(n,p^r))$$ is nilpotent.