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 S₄ and C₄, and the other made up from C_2h, C_2v 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

• Symmetry pages index
• Categorisation by symmetry operations or by unit cell parameters
• Correlation table index
• 3D objects
• Stereographs
• A brief description of tables

Last modified: Wed Nov 22 12:37:49 GMT 2000 by JPG (goss@excc.ex.ac.uk)
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