On nilpotent and solvable pair of polygroups

Document Type : Original Article


1 Department of Mathematics, Faculty of Science, Payame Noor University, Tehran, Iran

2 Department of Mathematics, Faculty of Science, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran


In this paper, we introduce nilpotent and solvable pair of polygroups. By a pair of polygroups $(P,N)$, we mean a polygroup $P$ with a normal sub-polygroup $N$ of $P$. In addition, we obtain a necessary and sufficient condition between nilpotency (solvability) of pair of polygroups and pair of groups. In particular, we extend the theory of groups to a pair of polygroups. Finally, we study the relationship between nilpotent and solvable pair of polygroups.


[1] R. Ameri, On categories of hypergroups and hypermodules, Journal of Discrete Mathematical
Sciences and Cryptography, 6 (2003), 121{132.
[2] R. Ameri, E. Mohammadzadeh, Engel groups derived from hypergroups, European Journal of
Cambinatorics, 44 (2015), 191{197.
[3] S.D. Comer, Extention of polygroups by polygroups and their representations using colour
schemes, Lecture Notes in Meth, No 1004, Universal Algebra and Lattice Theory, (1982),
[4] S.D. Comer, Hyperstructures assosiated with caracter algebras and colour schemes, in New
Ferontiers in Hyperstructures, Hadronic press, palm Harbor, FL, (1995), 49{66.
[5] P. Corsini, V. Leoreanu, Applications of hyperstructure theory, Kluwer Academic Publishers,
Dordrecht, 2003.
[6] P. Corsini, Prolegomena of hypergroup theory, Aviani Editore, Tricesimo, 1993.
[7] B. Davvaz, Polygroup theory and related systems, World Scienti c, 2013.
[8] G. Ellis, Capability, homology and central series of a pair of groups, Journal of Algebra, 176
(1996), 31{46.
[9] G. Ellis, The schur multiplier of a pair of groups, Applied Categorical Stractures, 6 (1998),
[10] D. Freni, Une note sur le cuur dun hypergroup et sur la cloture transitive β∗ de β (A note on
the core of a hypergroup and the transitivitive closure β∗ of β), Annali di Matematica Pura
ed Applicata, 8 (1991), 153{156 (in French).
[11] D. Freni, A new charactrization of the drived hypergroup via strongly regular equivalences,
Communications in Algebra, 30(8) (2002), 3977{3989.
[12] M. Hassanzadeh, A. Pormirzaei, S. Kayvanfar, On the nilpotency of pair of groups, Southeast
Asian Bulletin of Mathematics, 37 (2013), 67{77.
[13] T.W. Hungerford, Graduate tex in mathematics, Springer-Verlag New York, Heidelbery Berlin,
[14] M. Jafarpour, H. Aghabozorgi, B. Davvaz, On nilpotent and solvable polygroups, Bulletin of
the Iranian Mathematical Society, 39(3) (2013), 487{499.
[15] M. Koskas, Groupoides, demi-hypergroups ehypergroupes, Journal de Mathe'matiques Pures
et Applique'es, 49 (1970), 155{192.
[16] F. Marty, Sur une Generalization de la Notion de Groupe, in: 8th Congress Mathematics
Scandenaves, Stockholm, Sweden, (1934), 45{49.