N-central group

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Definition

Suppose $G$ is a group and $n$ is an integer other than 0. We say that $G$ is $n$-central if the exponent of the inner automorphism group of $G$ divides $n$.

Note that this is distinct from the notion of p-central group.

Facts

• If $n$ is an odd prime number $p$, then $n$-central implies $n$-abelian: see Frattini-in-center odd-order p-group implies p-power map is endomorphism
• n-abelian implies n(n-1)-central