Abstract
Hypergroups can be subdivided into two large classes: those whose heart coincide with the entire hypergroup and those in which the heart is a proper sub-hypergroup. The latter class includes the family of 1-hypergroups, whose heart reduces to a singleton, and therefore is the trivial group. However, very little is known about hypergroups that are neither 1-hypergroups nor belong to the first class. The goal of this work is to take a first step in classifying G-hypergroups, that is, hypergroups whose heart is a nontrivial group. We introduce their main properties, with an emphasis on G-hypergroups whose the heart is a torsion group. We analyze the main properties of the stabilizers of group actions of the heart, which play an important role in the construction of multiplicative tables of G-hypergroups. Based on these results, we characterize the G-hypergroups that are of type U on the right or cogroups on the right. Finally, we present the hyperproduct tables of all G-hypergroups of size not larger than 5, apart of isomorphisms.
Highlights
Hypercompositional algebra is a branch of Algebra that falls under the many generalizations of group theory [1]
The class of D-hypergroups consists of those hypergroups that are isomorphic to the quotient set of a group with respect to a non-normal subgroup, and is a subclass of cogroups [2,3,4], and cogroups appear as generalizations of C-hypergroups, that were introduced as hyperstructures having an identity element and a weak form of the cancellation law [5,6]
A strong link between group theory and hypergroup theory is established by the relation β, which is the smallest equivalence relation defined on a hypergroup H such that the corresponding quotient set H/β is a group [7,8,9]
Summary
Hypercompositional algebra is a branch of Algebra that falls under the many generalizations of group theory [1]. In [15,16], the authors characterized 1-hypergroups in terms of the height of their heart and provided a classification of the 1-hypergroups with |H| ≤ 6 based on the partition of H induced by β By means of this technique, the authors were able to enumerate all 1-hypergroups of size up to 6 and construct explicitly all non-isomorphic 1-hypergroups of size up to 5. These stabilizers play an important role in the construction of multiplicative tables of G-hypergroups, as they fix the hyperproducts g ◦ x and x ◦ g for all g ∈ ωH and x ∈ H. All the multiplicative tables of these hypergroups are listed and, using the results on 1-hypergroups found in [16], we conclude that there are 48 non-isomorphic G-hypergroups of size ≤ 5
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