✍ Scribed by Milan Randić
No coin nor oath required. For personal study only.
Structures X y 3 X Y * X Y g of symmetry (22" (of which propane is an example) are examined and the rearrangement due to the internal rotation of the end groups X Y s studied. The isomerization graph is constructed, various forms of which are displayed and the symmetry of which has been determined.
The order of the group is 72. There are nine prime (irreducible) representations (4A + E + 4G) with the following partitioning of the elements into classes: 1, 42, 62, 9, 122, 18. When the mechanism for rearrangement is generalized to include enantiomers, a duplex graph is produced with the order of the group 144 which is isomorphic to the group S2(Sg,S2) (generalized wreath product of the symmetric group S2 and 5'3). The corresponding graph has been constructed and displayed in one of more symmetrical forms. Isomorphism of groups of order 144 is discussed and a procedure is outlined in which correspondence between distinctive combinatorial objects is established by inducing permutations of m elements from available permutations of n elements. The scheme is based on selection of suitable graph invariants in one system and their labeling as m objects which form the basis for representation of the symmetry for the other system.
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