 Characteristic subgroup

A subgroup is characteristic when every automorphism of the ambient group maps it to itself. A uniquely determined subgroup of a given order or structural type is characteristic.

A **characteristic subgroup** of a group \( G \) is a subgroup \( H \) that is invariant under all automorphisms of the group \( G \). This means that for any automorphism \( \phi \) of \( G \), the image \( \phi(H) \) is still a subgroup of \( G \) and is equal to \( H \) itself.