# Conjugate subgroups

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*This article defines an equivalence relation over the collection of subgroups within the same big group*

## Contents

## Definition

### Symbol-free definition

Two subgroups of a group are termed **conjugate subgroups** if the following equivalent conditions are satisfied:

- There is an inner automorphism of the group that maps one subgroup bijectively to the other.
- They are in the same orbit under the group's action on its subgroup via inner automorphisms
- There is an action of the group on some set, where the two subgroups occur as isotropy subgroups of points in the same orbit.

### Definition with symbols

Two subgroups and of a group are termed **conjugate subgroups** if there is a in such that . Note that *exact equality* must hold.

### Why it is an equivalence relation

If we use the first definition, we need to justify as follows:

- Reflexivity: Because the identity map is an inner automorphism
- Symmetry: Because the inverse of an inner automorphism is also an inner automorphism
- Transitivity: Because the composite of inner automorphisms is an inner automorphism

The second definition makes it more or less obvious that it is an equivalence relation.

## Relation with other equivalence relations

### Stronger relations

### Weaker relations

- Automorphic subgroups
- Elementarily equivalently embedded subgroups
- Isomorphic subgroups: Two subgroups that are isomorphic as groups.