N-abelian group

This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property

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 $G$.

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.

Facts

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

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)

Relation with other properties

Weaker properties

• n-nilpotent group
• n-solvable group

Examples

Finite groups

We list examples of $n$-abelian finite groups for $n$ a positive integer, hence these also give examples of $(1-n)$-abelian groups by n-abelian iff (1-n)-abelian.

We list non-abelian examples of finite groups here only, all the abelian finite groups are trivially $n$-abelian for any given $n$.

$n$	Non-abelian $n$-abelian groups.
1	all non-abelian finite groups
2	no non-abelian finite groups (2-abelian iff abelian)
3	There are 10 3-abelian non-abelian finite groups with order at most 100 up to isomorphism: SmallGroup(27,3), SmallGroup(27,4), SmallGroup(54,10), SmallGroup(54,11), SmallGroup(81,3), SmallGroup(81,4), SmallGroup(81,6), SmallGroup(81,12), SmallGroup(81,13), SmallGroup(81,14).
4	There are 231 4-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest examples are dihedral group:D8 and quaternion group.
5	There are 221 5-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest examples are dihedral group:D8 and quaternion group.
6	There are 86 6-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest example is symmetric group:S3.