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Abstract
The concept of mark of a permutation group was introduced by Burnside in 1911. Redfield rediscovered marks in 1930s but his work was not published until 1984. He used marks to count group reduced distributions according to their symmetry groups. Marks have been used by several other authors; Sheehan [10] used marks to count the number of graphs with a given automorphism group, Lloyd [7] counted isomers using marks, Kamuti [4] used marks to find subdegrees of primitive permutation representations of PGL (2, q), Kimani et. al. [5] used marks to compute the ranks and subdegrees of the symmetric group acting on ordered pairs and ordered triples while Rimberia and Kamuti [9] used marks to find the subdegrees of primitive permutation representations of the symmetric group In this paper, we enumerate the number of stereoisomers derived from trigonal bipyramidal molecules using the mark version of Redfield superposition theorem. Results show that Iodotricarbonyl (triphenylphoshine) – cobalt (1) complex has 4 stereoisomers while a trigonal bipyramidal complex of the type MABCDE has 20 stereoisomers.
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References
Burnside, W. (1911). Theory of groups of finite order, Cambridge University Press, Cambridge (Dover reprint 1955). ISBN-10: 0486600386, ISBN-13: 978-0486600383.
Ivanov, A. A., Klin, M. H., Tsaranov, S. V. and Shpektorov, S.V. (1983). On the problem of computing subdegrees of transitive permutation groups. Soviet Mathematical Survey; 38: 123 – 124. doi:
https://doi.org/10.1070/RM1983v038n06ABEH003460
Kamuti, I. N. (1992). Combinatorial formulas, invariants and structures associated with primitive permutation representations of PSL (2, q) and PGL (2, q), Ph. D. Thesis, Southampton University, U.K.
Kamuti, I. N. (2006). Subdegrees of primitive permutation representations of PGL (2, q). East African Journal of Physical Sciences; 7(1/2): 25 – 41.
Kimani P., Rimberia J., Muthoka G., Lao H. (2014). Applications of marks to computation of ranks and subdegrees of the symmetric group acting on ordered pairs and ordered triples. Journal of Mathematical Theory and Modeling; 4:13 95-102. doi: https://doi.org/10.7176/MTM
Lloyd, E. K. (A). (1988). Redfield’s papers and their relevance to counting isomers and isomerizations, Discrete Appl. Math. 19: 289 – 304. doi: https://doi.org/10.1016/0166-218X(88)90020-0
Lloyd, E. K. (B). (1992). Marks of permutation groups and isomer enumeration. Journal of Mathematical Chemistry; 11 (1): 207 – 222. doi: https://doi.org/10.1007/BF01164205
Redfield, J. H. (1984). Enumeration by frame group and range groups, J. Graph Theory; 8:205 – 233.
Rimberia, J. K and Kamuti, I. N. (2015). Subdegrees of Primitive permutation representations of the Symmetric Group , International Journal of Scientific Research and Innovative Technology; 2: 4, 26-33. ISSN: 2313-3759 (Online).
Sheehan, J. (1968). The number of graphs with a given automorphism group, Canad. J. Math. 20: 1068 – 1076. doi: https://doi.org/10.1090/S0002-9939-1975- 0349408-9
White D. E. (1975). Counting patterns with a given automorphism group, Proc. Of Amer. Soc. 47:41 – 44. doi: https://doi.org/10.1090/S0002-9939-75-99973-6