Journal of the
Korean Mathematical Society

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008



J. Korean Math. Soc. 2024; 61(1): 109-132

Online first article December 21, 2023      Printed January 1, 2024

Copyright © The Korean Mathematical Society.

Combinatorial supersymmetry: Supergroups, superquasigroups, and their multiplication groups

Bokhee Im, Jonathan D.H. Smith

Chonnam National University; Iowa State University


The Clifford algebra of a direct sum of real quadratic spaces appears as the superalgebra tensor product of the Clifford algebras of the summands. The purpose of the current paper is to present a purely set-theoretical version of the superalgebra tensor product which will be applicable equally to groups or to their non-associative analogues --- quasigroups and loops. Our work is part of a project to make supersymmetry an effective tool for the study of combinatorial structures. Starting from group and quasigroup structures on four-element supersets, our superproduct unifies the construction of the eight-element quaternion and dihedral groups, further leading to a loop structure which hybridizes the two groups. All three of these loops share the same character table.

Keywords: Quasigroup, loop, superalgebra, character table

MSC numbers: 20N05, 17A70, 20C15

Supported by: The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (NRF-2017R1D1A3B05029924).

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