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.

Relation with other equivalence relations

Stronger relations

• Equal subgroups

Weaker relations

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