Conjugate-coset space

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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.