Symmetry, the orientational automorphism group

Note that there is no 1-1-correspondence between orientation and the Cahn-ln-gold-Prelog (CIP) convention. The orientation of a particular quadruple depends on the numbering of atoms, not on their nature. Thus, exchanging atom numbers 3 and 4 in conformation 4.2a, as shown in 4.2a , will result in the orientation 

6 Remark (A test on chirality) Orientation functions obtained from each other by a graph-automorphic renumbering are isomorphic. A conformation is achiral if and only if its orientation function is isomorphic to its negative. 

For example, the fact that the orientation function of the cyclohexane conformer is isomorphic to its negative indicates the achirality of the conformer. 

Note that achirality does not imply a balanced relationship between plus and minus signs in the orientation function. Note also that the exchange of atom numbers 3 

Isomorphic orientation functions differ only in the chosen atom numbering, they belong to essentially equivalent conformations. Thus, when generating conformations we would like to generate exactly one representative from each class of isomorphic orientation functions. This is achieved by defining canonic orientation functions. Within each class of isomorphic orientation functions, we declare one as canonic. Thus, an efficient test to recognize whether a given orientation function is canonic or not is necessary. 

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