N-abelian group
From Groupprops
This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property
Contents
Definition
Suppose is an integer. A group
is termed a
-abelian group if the
power map
is an endomorphism of
, i.e.,
for all
. If this is the case, then the
power map is termed a universal power endomorphism of
.
As noted below, n-abelian iff (1-n)-abelian, so it suffices to restrict attention to 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
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
Particular values
Value of ![]() ![]() ![]() |
Characterization of ![]() |
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) |