IA not implies class-preserving
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)
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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 is a perfect group, then every automorphism of it is an IA-automorphism, so all we need to do is exhibit an automorphism of that is not class-preserving. Indeed, take to be the alternating group of degree five, acting on . Consider the automorphism of induced via conjugation by any odd permutation. We claim that such an automorphism sends the five-cycle to an element outside its conjugacy class.
Suppose not. Then, there exists an even permutation that has the same effect by conjugation on . Taking their ratio, we get an odd permutation that commutes with . But we know that the only permutations that commute with are its powers, which are all even permutations -- hence a contradiction.
A more generic way of saying this is that if is a perfect centerless group that is not conjugacy-closed in the automorphism group, then has IA-automorphisms that are not class-preserving.