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Abstract
Let G be a finite group and \hat{G} be the set of all irreducible complex characters of G. In this paper, we consider <\hat{G}, *> as a polygroup, where for each chi_i ,chi_j in \hat{G} the product \chi _{i} * \chi_{j} is the set of those irreducible constituents which appear in the element-wise product \chi_{i} \chi_{j}. We call that \hat{G} simple if it has no proper normal subpolygroup and show that if \hat{G} is a single power cyclic polygroup, then \hat{G} is a simple polygroup and hence \hat{S}_{n} and \hat{A}_{n} are simple polygroups. Also, we prove that if G is a non-abelian simple group, then \hat{G} is a single power cyclic polygroup. Moreover, we classify \hat{D}_{2n} for all n. Also, we prove that \hat{T}_{4n} and \hat{U}_{6n} are cyclic polygroups with finite period.
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References
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References
Brauer, R. "{On pseudo groups}", {em J. Math. Soc. Japan}. 20 (1968),
-22.
Comer, S.D., "Hyperstructures associated with character algebra and color schemes", {em New Frontiers in Hyperstructures}, Hadronic Press, (1996) 49-66.
Corsini, P., {em Prolegomena of Hypergroup Theory}, ( Aviani Editore, Tricesimo), (1993).
Davvaz, B., {em Polygroup Theory and Related Systems}, World Scientific Publishing co. Pte. Ltd, (2013).
James. G., Liebeck. M, {em Representations and characters of groups}, Cambridge Univ. Press, (1993).
Konguetsof, L., Vougiouklis, T., Kessoglides, M. and Spartalis, S., "{ On cyclic hypergroups with period}", {em Acta Univ. Carolinae-Math. Physica.} 28 (1987), no 1, 3-7.
Leoreanu, V., "{ About the simplifiable cyclic semihypergroups}", {em Italian J. Pure Appl. Math.} 7 (2000), 69-76.
Mittas, J., "Hypergroupes canoniques hypervalues",(French), {em C. R. Acad. Sci. Paris Ser, A-B}, 271 (1970), A4- A-7.
Roth, R.L., "{ Character and conjugacy class hypergroups of a finite group}", {em Ann. Math. Pura Appl.} 105 (1975), 295-311.
Sekhavatizadeh, S., Zahedi, M.M. and Iranmanesh, A., "{ Cyclic hypergroups which are induced by the character of some finite groups}", {em Italian j. pure appl.} 33 (2014), 123-132.
Vougiouklis, T., "{ Cyclicity in a special class of hypergroups}", {em Acta Univ. Carolinae-Math.}
et Physica. 22 (1981), 3-6.
Wall, H.S., "{ Hypergroups}", {em Amer. J. Math.} 59 (1937), 77-98.