Combinatorial Properties, Invariants and Structures of the Action of Sn×An on X×Y

Authors

  • Amos Kituku Mutua Amos Kituku Mutua, P.O Box 62000, Juja-00200, Kenya
  • Lewis Namu Nyaga Lewis Namu Nyaga, P.O Box 62000, Juja-00200, Kenya
  • Richard Kiriuki Gachimu Richard Kariuki Gachimu, P.O Box 62000, Juja-00200, Kenya

Keywords:

Direct Product, Symmetric Group, Alternating Group, Action, Rank, Subdegrees, Suborbital.

Abstract

The transitivity, primitivity, rank and subdegrees, as well as pairing of the suborbits associated with the action of the actions of the direct product , of the symmetric group    by the alternating group   alternating on the Cartesian product , where  and  are disjoint sets each containing n elements is an area that has never received attention from researchers for a very long time. In this paper, we prove that the action is both transitive and imprimitive when . Also, we establish that that the rank is  if , but is  for all . In addition, we show in this paper that the subdegrees associated with the action are . Lastly, we show that all the suborbits corresponding to the action, are self-paired when  

References

Sims, C. C. (1967). Graphs and finite permutation groups. Mathematische Zeitschrift, 95:76–86.

Wielandt, H. (1964). Finite Permutation Groups. Academic Press New York.

D. G. Higman, Characterization of Families of Rank 3 Permutation Groups by the Subdegrees I, Arch. Math., 21, No. 1 (1970), 151–156.

Rose, J. S. (1978). A Course in Group Theory. Cambridge,University Press, Cambridge.

C. F. Gardiner, Algebraic Structures, Ellis Horwood, England (1986).

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Published

2018-08-11

How to Cite

Mutua, A. K., Nyaga, L. N., & Gachimu, R. K. (2018). Combinatorial Properties, Invariants and Structures of the Action of Sn×An on X×Y. International Journal of Sciences: Basic and Applied Research (IJSBAR), 40(2), 109–115. Retrieved from https://www.gssrr.org/index.php/JournalOfBasicAndApplied/article/view/9114

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