N-abelian group

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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^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.

Facts

General facts


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