Conjugate-coset space

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 $$G/H \times G/H \to CU(G,H)$$. Any automorphism will act separately on each coordinate $$G/H$$, and will also preserve the mapping. This explains both the above observations.

Particular cases
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 $$G/H$$, the usual coset space. Further, it is equipped with a group structure, and the mapping $$G/H \times G/H \to CU(G,H)$$ is the group multiplication in this group structure.
 * Self-normalizing subgroup: When the subgroup is self-normalizing, the map $$G/H \times G/H \to CU(G,H)$$ is bijective.