# 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 Cn 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 S4 and C4, and the other made up from C2h, C2v and D2.

 Order Sn Cn Cnh Cnv Dn Dnd Dnh Cubic Icosahedral 1 C1 2 S2 C2 C1h 3 C3 4 S4 C4 C2h C2v D2 5 C5 6 S6 C6 C3h C3v D3 7 C7 8 S8 C8 C4h C4v D4 D2d D2h 9 C9 10 S10 C10 C5h C5v D5 12 S12 C6h C6v D6 D3d D3h T 14 S14 C7h C7v D7 16 S16 C8h C8v D8 D4d D4h 18 S18 C9h C9v D9 20 S20 C10h C10v D10 D5d D5h 24 D6d D6h O, Td Th 48 Oh 60 I 120 Ih