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., $$(xy)^n = x^ny^n$$ for all $$x,y \in G$$. If this is the case, then the $$n^{th}$$ power map is termed a universal power endomorphism of $$G$$.

As noted below, n-abelian iff (1-n)-abelian, so it suffices to restrict attention to $$n$$ a positive integer.

Alternative definitions
See Alperin's structure theorem for n-abelian groups.

General facts

 * n-abelian iff (1-n)-abelian
 * 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.
 * abelian implies n-abelian for all n
 * n-abelian implies every nth power and (n-1)th power commute
 * n-abelian implies n(n-1)-central
 * n-abelian iff abelian (if order is relatively prime to n(n-1))
 * nth power map is surjective endomorphism implies (n-1)th power map is endomorphism taking values in the center
 * (n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism
 * Frattini-in-center odd-order p-group implies p-power map is endomorphism
 * Frattini-in-center odd-order p-group implies (mp plus 1)-power map is automorphism
 * Characterization of exponent semigroup of a finite p-group
 * Alperin's structure theorem for n-abelian groups

Particular values
 

Weaker properties

 * n-nilpotent group
 * n-solvable group