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G -Hypergroups: Hypergroups with a Group-Isomorphic Heart

Author

Listed:
  • Mario De Salvo

    (Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università di Messina, 98122 Messina, Italy)

  • Dario Fasino

    (Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, 33100 Udine, Italy)

  • Domenico Freni

    (Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, 33100 Udine, Italy)

  • Giovanni Lo Faro

    (Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università di Messina, 98122 Messina, Italy)

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.

Suggested Citation

  • Mario De Salvo & Dario Fasino & Domenico Freni & Giovanni Lo Faro, 2022. "G -Hypergroups: Hypergroups with a Group-Isomorphic Heart," Mathematics, MDPI, vol. 10(2), pages 1-17, January.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:2:p:240-:d:724001
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