Point Groups in Three Dimensions - Correspondence Between Rotation Groups and Other Groups

Correspondence Between Rotation Groups and Other Groups

The following groups contain inversion:

• C_nh and D_nh for even n
• S_2n and D_nd for odd n (S₂ = C_i is the group generated by inversion; D_1d = C_2h)
• T_h, O_h, and I_h

As explained above, there is a 1-to-1 correspondence between these groups and all rotation groups:

• C_nh for even n and S_2n for odd n correspond to C_n
• D_nh for even n and D_nd for odd n correspond to D_n
• T_h, O_h, and I_h correspond to T, O, and I, respectively.

The other groups contain indirect isometries, but not inversion:

• C_nv
• C_nh and D_nh for odd n
• S_2n and D_nd for even n
• T_d

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:

• C_n is subgroup of D_n of index 2, giving C_nv
• C_n is subgroup of C_2n of index 2, giving C_nh for odd n and S_2n for even n
• D_n is subgroup of D_2n of index 2, giving D_nh for odd n and D_nd for even n
• T is subgroup of O of index 2, giving T_d