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. 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. Prime power order implies nilpotent
3. Equivalence of definitions of periodic nilpotent group: The key thing we use is that any finitely generated periodic nilpotent group is finite.
4. 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.

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
1	Any finite quotient group of $B(n,p^{r})$ is a finite $p$-group, and hence nilpotent.	Facts (1), (2)			finite group where, by definition, the order of every element divides $p^{r}$, hence is a power of $p$. By Fact (1), $p$ is the only prime that can divide the order of the quotient group. Hence the quotient group is a finite $p$-group. Thus, by Fact (2), it is nilpotent.
2	Any nilpotent quotient group of $B(n,p^{r})$ is finite.	Fact (3)			$B(n,p^{r})$ is finitely generated and has finite exponent, so any nilpotent quotient group of it is a finitely generated nilpotent group of finite exponent. Hence, by Fact (3), it must be finite.
3	The quotient group by a given normal subgroup of $B(n,p^{r})$ is finite if and only if it is nilpotent.			Steps (1), (2)	Step-combination direct.
4	The finite residual of $B(n,p^{r})$ is the same as the nilpotent residual of $B(n,p^{r})$. Thus, $RB(n,p^{r})$ is the quotient of $B(n,p^{r})$ by its nilpotent residual.			Step (3)	Follows from Step (3), and the observation that the finite residual and nilpotent residual are defined respectively as the intersections of normal subgroups with finite (respectively nilpotent) quotient groups.
5	$RB(n,p^{r})$ is finite if and only if it is nilpotent if and only if the quotient of $B(n,p^{r})$ by its nilpotent residual is nilpotent.			Steps (1), (4)	Use Step (1) to conclude that $RB(n,p^{r})$ is finite if and only if it is nilpotent. Now use Step (4) to note that it is the same as the quotient by the nilpotent residual.
6	$RB(n,p^{r})$ is finite if and only if $L(B(n,p^{r}))$ is nilpotent.	Fact (4)		Step (5)	Step-fact combination direct