# 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) neednotsatisfy the second automorphism property (i.e., class-preserving automorphism)

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## Contents

## Statement

### 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.

## Proof

### 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.