N-abelian group

Definition

Suppose $n$ is an integer. A group $G$ is termed a $n$-abelian group if the $n^{th}$ power map $x\mapsto x^n$ is an endomorphism of $G$, i.e., $(xy{)}^n=x^n{y}^n$ for all $x,y\in G$. If this is the case, then the $n^{th}$ power map is termed a universal power endomorphism of $G$.

The set of $n$ for which $G$ is $n$-abelian is termed the exponent semigroup of $G$. It is a submonoid of the multiplicative monoid of integers.

Facts

• Every group is 0-abelian and 1-abelian.
• Abelian implies n-abelian for all n
• 2-abelian iff abelian
• -1-abelian iff abelian
• n-abelian iff (1-n)-abelian
• n-abelian implies every nth power and (n-1)th power commute
• n-abelian implies n(n-1)-central

Relation with other properties

Weaker properties

• n-nilpotent group
• n-solvable group