# Conjugate subgroups

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

## 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 $H_1$ and $H_2$ of a group $G$ are termed conjugate subgroups if there is a $g$ in $G$ such that $gH_1g^{-1} = H_2$. 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.