An automorphism that sends any subgroup to a conjugate thereof is termed a subgroup-conjugating automorphisms. The group of subgroup-conjugating automorphisms acts naturally on the conjugate-coset space: it takes any coset of a subgroup to a coset of a conjugate subgroup. The action of the automorphism group on the conjugate-coset space has the following interesting features:
- It takes all left cosets of a single subgroup to left cosets of a single subgroup
- It takes all right cosets of a single subgroup to right cosets of a single subgroup
Thinking of it graphically, consider the map . Any automorphism will act separately on each coordinate , and will also preserve the mapping. This explains both the above observations.
There are some particular cases of the subgroup where this action becomes interesting:
- Normal subgroup: When the subgroup is normal, the conjugate-coset space is just , the usual coset space. Further, it is equipped with a group structure, and the mapping is the group multiplication in this group structure.
- Self-normalizing subgroup: When the subgroup is self-normalizing, the map is bijective.