Inverse map is automorphism iff abelian
This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself
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This article gives a result about how information about the structure of the automorphism group of a group (abstractly, or in action) can control the structure of the group
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The following are equivalent for a group:
- The map sending every element to its inverse, is an endomorphism
- The map sending every element to its inverse, is an automorphism
- The group is abelian
The equivalence of (1) and (2) is direct from the fact that the inverse map is bijective.
Similar facts for other power maps
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)|
- Fixed-point-free involution on finite group is inverse map
- Automorphism sends more than three-fourths of elements to inverses implies abelian
Related facts for other algebraic structures
- Multiplication by n map is an endomorphism iff derived subring has exponent dividing n(n-1) for Lie rings. In particular, we can construct examples of non-abelian Lie rings where the negation map is an automorphism.
- Inverse map is automorphism not implies abelian for loop
Given: A group
To prove: is Abelian iff the map is an automorphism.
Proof: The following fact is true:
Thus, we see that:
Since the inverse map is a bijection, this tells us that the above is a homomorphism iff any two elements commute.