Inverse map is automorphism iff abelian

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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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Statement

The following are equivalent for a group:

1. The map sending every element to its inverse, is an endomorphism
2. The map sending every element to its inverse, is an automorphism
3. The group is abelian

The equivalence of (1) and (2) is direct from the fact that the inverse map is bijective.

Related facts

Similar facts for other power maps

We say that a group is a n-abelian group if the $n^{th}$ power map is an endomorphism. Here are some related facts about $n$-abelian groups.

• n-abelian iff (1-n)-abelian
• The set of $n$ for which $G$ is $n$-abelian is termed the exponent semigroup of $G$. 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 $n$ (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)

Applications

• 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

Proof

Given: A group $G$

To prove: $G$ is Abelian iff the map $x\mapsto x^{-1}$ is an automorphism.

Proof: The following fact is true:

$(xy{)}^{-1}=y^{-1}{x}^{-1}$

Thus, we see that:

$(xy{)}^{-1}=x^{-1}{y}^{-1}⟺x^{-1},y^{-1}$ commute

Since the inverse map is a bijection, this tells us that the above is a homomorphism iff any two elements commute.

Textbook references

• Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 71, Exercise 12(b) of Section 3 (Isomorphisms)