# Automorphism that preserves conjugacy classes for a generating set

From Groupprops

This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)

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## Contents

## Definition

Suppose is a group and is an automorphism of .We say that is an **automorphism that preserves conjugacy classes for a generating set** if the folllowing equivalent conditions hold:

- There is a generating set for such that, for all , is in the conjugacy class of , i.e., there exists such that . Note that depends on .
- There is a symmetric subset of that is a generating set of and such that, for all , is in the conjugacy class of , i.e., there exists such that . Note that depends on .
- There is a normal subset of that is a generating set of and such that, for all , is in the conjugacy class of , i.e., there exists such that . Note that depends on .

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

inner automorphism | an automorphism for which there exists a single element such that the automorphism is the map | (obvious) | (via class-preserving automorphism) | |FULL LIST, MORE INFO |

locally inner automorphism | looks like an inner automorphism in its effect on any finite subset | (obvious) | (via class-preserving automorphism) | |FULL LIST, MORE INFO |

class-preserving automorphism | sends every element to within its conjugacy class | (obvious) | preserves conjugacy classes for a generating set not implies class-preserving | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

IA-automorphism | sends every element to within its coset of the derived subgroup, or equivalently, induces the identity automorphism on the abelianization | preserves conjugacy classes for a generating set implies IA | IA-automorphism|automorphism that preserves conjugacy classes for a generating set]] |