N-abelian iff abelian (if order is relatively prime to n(n-1))
This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself
View other elementary non-basic facts
VIEW FACTS USING THIS: directly | directly or indirectly, upto two steps | directly or indirectly, upto three steps|
VIEW: Survey articles about this
Statement
Suppose is a finite group and
is an integer such that the order of
is relatively prime to
. Then, the
power map on
is an endomorphism (i.e.,
is a n-abelian group if and only if
is an abelian group.
Related facts
Facts about n-abelian groups
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 ![]() ![]() ![]() |
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) |
Analogues in other algebraic structures
- Multiplication by n map is a derivation iff derived subring has exponent dividing n
- Multiplication by n map is an endomorphism iff derived subring has exponent dividing n(n-1)
Tightness
Further information: Frattini-in-center odd-order p-group implies p-power map is endomorphism, Frattini-in-center odd-order p-group implies (p plus 1)-power map is automorphism
It turns out that the condition of being relatively prime to is fairly tight for
, i.e., we can find non-abelian groups of order not relatively prime to
, as well as non-abelian groups of order not relatively prime to
, where the
power map is an endomorphism.
- The
case:
must be divisible either by 4 or by an odd prime. If 4 divides
, the
power map gives an automorphism for any non-abelian group of exponent 4, such as dihedral group:D8. If
is an odd prime dividing
, we can find a non-abelian Frattini-in-center
-group, where the
power map is an endomorphism taking values in the center. Hence, the
power map is an automorphism.
- The
case:
must be divisible either by 4 or by an odd prime. If 4 divides
, the
power map is an endomorphism of any non-abelian group of exponent 4. If
is an odd prime dividing
, we can find a non-abelian Frattini-in-center
-group, where the
power map is an endomorphism taking values in the center. Hence, the
power map is an endomorphism.
Facts used
- Abelian implies universal power map is endomorphism
- nth power map is endomorphism implies every nth power and (n-1)th power commute
- kth power map is bijective iff k is relatively prime to the order
Proof
From abelianness to the power map being an endomorphism
This follows from fact (1).
From the power map being an endomorphism to being Abelian
Given: A finite group and an integer
such that the order of
is relatively prime to
. Further,
is an endomorphism of
.
To prove: is abelian.
Proof:
Step no. | Assertion | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | We get ![]() ![]() ![]() ![]() |
Fact (2) | ![]() |
Given+Fact direct | |
2 | ![]() |
Fact (3) | Step (1) | [SHOW MORE] |