Universal power endomorphism

This article defines a function property, viz a property of functions from a group to itself

Definition

An endomorphism of a group is termed a universal power endomorphism if there exists an integer $n$ such that the endomorphism can be expressed as $g\mapsto g^{n}$ for all $g$ in the group.

Note that for $n=0,1$, the $n^{th}$ power map is a universal power endomorphism. For other $n$, it is a universal power endomorphism if the group is abelian, otherwise it need not be.

If, for a particular group $G$, the $n^{th}$ power map is an endomorphism, then we may say that $G$ is a $n$-abelian group (see n-abelian group).

Facts

We say that a group is a n-abelian group if the $n^{th}$ power map is an endomorphism. Here are some related facts about $n$-abelian groups.

• 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

Value of $n$ (note that the condition for $n$ is the same as the condition for $1-n$)	Characterization of $n$-abelian groups	Proof	Other related facts
0	all groups	obvious
1	all groups	obvious
2	abelian groups only	2-abelian iff abelian	endomorphism sends more than three-fourths of elements to squares implies abelian
-1	abelian groups only	-1-abelian iff abelian
3	3-abelian group means: 2-Engel group and derived subgroup has exponent dividing three	Levi's characterization of 3-abelian groups	cube map is surjective endomorphism implies abelian, cube map is endomorphism iff abelian (if order is not a multiple of 3), cube map is endomorphism implies class three
-2	same as for 3-abelian	(based on n-abelian iff (1-n)-abelian)