N-local nilpotency class n implies nilpotency class n

Statement

For Lie rings

Suppose $L$ is a Lie ring and $n$ is a natural number such that $n\geq 3$ and the $n$-local nilpotency class of $L$ is at most $n$. Then, $L$ is a nilpotent Lie ring and the nilpotency class of $L$ is at most $n$.

Note that in the case that $n=2$, $L$ need not be a Lie ring of nilpotency class two. However, we still must have that $L$ is a Lie ring of nilpotency class three.

For groups

Suppose $G$ is a group and $n$ is a natural number such that $n\geq 3$ and the $n$-local nilpotency class of $G$ is at most $n$. Then, $G$ is a nilpotent group and the nilpotency class of $G$ is at most $n$.

Note that in the case that $n=2$, $G$ need not be a group of nilpotency class two. However, we still must have that $G$ is a group of nilpotency class three.

Related facts

• 2-Engel implies class three for Lie rings, 2-Engel implies class three for groups (this is the $n=2$ case)

Facts used

1. Nilpotency class three is 3-local for Lie rings, nilpotency class three is 3-local for groups
2. Relation between local nilpotency class of Lie ring and inner derivation ring, relation between local nilpotency class of group and inner automorphism group,

Proof

Proof for Lie rings

Given: A Lie ring $L$, $n\geq 3$, the $n$-local nilpotency class of $L$ is at most $n$.

To prove: The nilpotency class of $L$ is at most $n$.

Proof: We use Fact (2) to show that, if we take the quotient of $L$ by the $(n-3)^{th}$ member of its Lie ring, i.e., we consider the quotient $L/Z^{(n-3)}(L)$, this is a Lie ring whose 3-local nilpotency class is at most 3. Now, we use Fact (1) to show that $L/Z^{(n-3)}(L)$ has class at most three. Hence, $L$ has class at most $n$.

Proof for groups

Given: A group $G$, $n\geq 3$, the $n$-local nilpotency class of $G$ is at most $n$.

To prove: The nilpotency class of $G$ is at most $n$.

Proof: We use Fact (2) to show that, if we take the quotient of $G$ by the $(n-3)^{th}$ member of its upper central series, i.e., we consider the quotient $G/Z^{(n-3)}(G)$, this is a group whose 3-local nilpotency class is at most 3. Now, we use Fact (1) to show that $G/Z^{(n-3)}(G)$ has class at most three. Hence, $G$ has class at most $n$.