Relation between local nilpotency class of group and inner automorphism group

Statement

Suppose $G$ is a group and $m,c$ are positive integers such that the $m$-local nilpotency class of $G$ is at most $c$. Suppose $Z(G)$ is the center of $G$, so that the inner automorphism group of $G$, denoted $\operatorname {Inn} (G)$, is isomorphic to $G/Z(G)$. Then, the $(m-1)$-local nilpotency class of $\operatorname {Inn} (G)$ is at most $c-1$.

Note that if we took $m=\infty$, this would simply tell us that the nilpotency class of the inner automorphism group is one less than that of the original group.

Related facts

• Relation between local nilpotency class of group and derived subgroup
• Relation between local nilpotency class of Lie ring and inner derivation ring