Correspondence Between Rotation Groups and Other Groups
The following groups contain inversion:
- Cnh and Dnh for even n
- S2n and Dnd for odd n (S2 = Ci is the group generated by inversion; D1d = C2h)
- Th, Oh, and Ih
As explained above, there is a 1-to-1 correspondence between these groups and all rotation groups:
- Cnh for even n and S2n for odd n correspond to Cn
- Dnh for even n and Dnd for odd n correspond to Dn
- Th, Oh, and Ih correspond to T, O, and I, respectively.
The other groups contain indirect isometries, but not inversion:
- Cnv
- Cnh and Dnh for odd n
- S2n and Dnd for even n
- Td
They all correspond to a rotation group H and a subgroup L of index 2 in the sense that they are obtained from H by inverting the isometries in H \ L, as explained above:
- Cn is subgroup of Dn of index 2, giving Cnv
- Cn is subgroup of C2n of index 2, giving Cnh for odd n and S2n for even n
- Dn is subgroup of D2n of index 2, giving Dnh for odd n and Dnd for even n
- T is subgroup of O of index 2, giving Td
Read more about this topic: Point Groups In Three Dimensions
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