Conjugate subgroups

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.

Stronger relations

 * Weaker than::Equal subgroups

Weaker relations

 * Stronger than::Automorphic subgroups
 * Stronger than::Elementarily equivalently embedded subgroups
 * Stronger than::Isomorphic subgroups: Two subgroups that are isomorphic as groups.