Universal power endomorphism
This article defines a function property, viz a property of functions from a group to itself
Note that for , the power map is a universal power endomorphism. For other , it is a universal power endomorphism if the group is abelian, otherwise it need not be.
If, for a particular group , the power map is an endomorphism, then we may say that is a -abelian group (see n-abelian group).
We say that a group is a n-abelian group if the power map is an endomorphism. Here are some related facts about -abelian groups.
- n-abelian iff (1-n)-abelian
- The set of for which is -abelian is termed the exponent semigroup of . 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 (note that the condition for is the same as the condition for )||Characterization of -abelian groups||Proof||Other related facts|
|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)|