Isomorphism |
Isomorphism is a mathematical equivalence between two or more groups. Isomorphic groups possess the same structure in the character tables, but differ in symmetry operations and selection rules.
Finite isomorphic groups must possess the same number of element of any given period. The period of an element is defined as the number of times that operation must be performed to be equivalent to the identity operation. For example, the C_{n} operation is period n and any mirror operation is period 2.
Collections of isomorphic groups are said to belong to the same abstract group, which do not in general correspond to crystallographic categories.
The following table [after I Novak, Eur. J. Phys. 16, 151 (1995)] lists many molecular point groups by order. Abstract groups are by definition must have the same order, and hence lie on the same row of the table. Abstract groups are grouped by the coloured backgrounds. For example, there are two abstract groups of order 4, one made up from S_{4} and C_{4}, and the other made up from C_{2h}, C_{2v} and D_{2}.
Order | S_{n} | C_{n} | C_{nh} | C_{nv} | D_{n} | D_{nd} | D_{nh} | Cubic | Icosahedral |
1 | C_{1} | ||||||||
2 | S_{2} | C_{2} | C_{1h} | ||||||
3 | C_{3} | ||||||||
4 | S_{4} | C_{4} | C_{2h} | C_{2v} | D_{2} | ||||
5 | C_{5} | ||||||||
6 | S_{6} | C_{6} | C_{3h} | C_{3v} | D_{3} | ||||
7 | C_{7} | ||||||||
8 | S_{8} | C_{8} | C_{4h} | C_{4v} | D_{4} | D_{2d} | D_{2h} | ||
9 | C_{9} | ||||||||
10 | S_{10} | C_{10} | C_{5h} | C_{5v} | D_{5} | ||||
12 | S_{12} | C_{6h} | C_{6v} | D_{6} | D_{3d} | D_{3h} | T | ||
14 | S_{14} | C_{7h} | C_{7v} | D_{7} | |||||
16 | S_{16} | C_{8h} | C_{8v} | D_{8} | D_{4d} | D_{4h} | |||
18 | S_{18} | C_{9h} | C_{9v} | D_{9} | |||||
20 | S_{20} | C_{10h} | C_{10v} | D_{10} | D_{5d} | D_{5h} | |||
24 | D_{6d} | D_{6h} | O, T_{d} | ||||||
T_{h} | |||||||||
48 | O_{h} | ||||||||
60 | I | ||||||||
120 | I_{h} |