Certain Cyclic Hypergroups with Property of Subhypergroup Indices

Abstract

Background and Objectives : Cyclic groups are one of the important groups which have been studied in many terms of characterizations. Property(D) of a group G is a property that distinct subgroups of G  have distinct indices in G.  Considering finite and infinite groups separately, it is proved that an arbitrary group is cyclic if and only if it has property(D). That means both groups (Z,+)  and (Z_n,+_n) have property(D). Also, hypergroups are generalizations of groups. In this research, we extend the study from groups to hypergroups in order to determine being cyclic hypergroups and property(D) of the hypergroups (Z,o_k) and (Z_n,o^'_k).   

Methodology : Using hyperoperations and partitions of sets Z and Z_n to determine being cyclic hypergroups. Finding all identity elements and all subhypergroups together with invertibilities to consider the index of each subhypergroup and prove having property(D) of both hypergroups.

Main Results : The hypergroups (Z,o_k) and (Z_n,o^'_k) are cyclic except k = 0 in (Z,o_k) and both hypergroups have property(D).

Conclusions  : In arbitrary hypergroups, property(D) is not a characterization of cyclic hypergroups, which is different from the results in groups. However, the indices of subhypergroups of (Z,o_k) and (Z_n,o^'_k)  have the same results as the indices of subgroups of (Z,+) and (Z_n,+_n)  respectively.