# Class-preserving implies IA

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., class-preserving automorphism) must also satisfy the second automorphism property (i.e., IA-automorphism)

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## Contents

## Statement

### Property-theoretic statement

The automorphism property of being a Class-preserving automorphism (?) is stronger than the automorphism property of being an IA-automorphism (?).

## Definitions used

### Class automorphism

`Further information: Class-preserving automorphism`

An automorphism of a group is termed a class automorphism if it sends every element to an element in its conjugacy class; in other words, it preserves conjugacy classes.

### IA-automorphism

`Further information: IA-automorphism`

An automorphism of a group is termed an IA-automorphism if it induces the identity map on the Abelianization of the group. In other words, it sends every coset of the commutator subgroup, to itself.