IA not implies class-preserving

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This article gives the statement and possibly, proof, of a non-implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., IA-automorphism) need not satisfy the second automorphism property (i.e., class-preserving automorphism)
View a complete list of automorphism property non-implications | View a complete list of automorphism property implications
Get more facts about IA-automorphism|Get more facts about class-preserving automorphism


Property-theoretic statement

The automorphism property of being an IA-automorphism does not imply the automorphism property of being a class-preserving automorphism.

Verbal statement

There exists a group with an IA-automorphism (an automorphism that is identity on the Abelianization) that does not send every element to within its conjugacy class.


An example of a perfect group

If G is a perfect group, then every automorphism of it is an IA-automorphism, so all we need to do is exhibit an automorphism of G that is not class-preserving. Indeed, take G to be the alternating group of degree five, acting on \{ 1,2,3,4,5\}. Consider the automorphism of G induced via conjugation by any odd permutation. We claim that such an automorphism sends the five-cycle (1,2,3,4,5) to an element outside its conjugacy class.

Suppose not. Then, there exists an even permutation that has the same effect by conjugation on (1,2,3,4,5). Taking their ratio, we get an odd permutation that commutes with (1,2,3,4,5). But we know that the only permutations that commute with (1,2,3,4,5) are its powers, which are all even permutations -- hence a contradiction.

A more generic way of saying this is that if G is a perfect centerless group that is not conjugacy-closed in the automorphism group, then G has IA-automorphisms that are not class-preserving.