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Designs, Codes and Cryptography

, Volume 48, Issue 1, pp 59–68 | Cite as

2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four

Article

Abstract

A ternary quasigroup (or 3-quasigroup) is a pair (N, q) where N is an n-set and q(x, y, z) is a ternary operation on N with unique solvability. A 3-quasigroup is called 2-idempotent if it satisfies the generalized idempotent law: q(x, x, y) = q(x, y, x) = q(y, x, x) = y. A conjugation of a 3-quasigroup, considered as an OA(3, 4, n), \({(N, \mathcal{B})}\) , is a permutation of the coordinate positions applied to the 4-tuples of \({\mathcal{B}}\) . The subgroup of conjugations under which \({(N, \mathcal{B})}\) is invariant is called the conjugate invariant subgroup of \({(N, \mathcal{B})}\). In this paper, we will complete the existence proof of the last undetermined infinite class of 2-idempotent 3-quasigroups of order n, n ≡ 1 (mod 4) and n > 9, with a conjugate invariant subgroup consisting of a single cycle of length four.

Keywords

3-quasigroup Orthogonal array Quadruple system 

AMS Classification

05B15 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsSuzhou UniversitySuzhouChina

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