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Parity of orthogonal automorphisms. (English) Zbl 0632.20041

Let Q(\(\cdot)\) be a quasigroup. For each \(a\in Q\) the equations \({\mathcal L}(a,Q)(x)=ax\) and \({\mathcal R}(a,Q)(x)=xa\) define the left and the right translations by a. Let \({\mathcal M}_{\ell}(Q)=\{{\mathcal L}(a,Q)\); \(a\in Q\}\), \({\mathcal M}_ r(Q)=\{{\mathcal R}(a,Q)\); \(a\in Q\}\) and \({\mathcal M}(Q)={\mathcal M}_{\ell}(Q)\cup {\mathcal M}_ r(Q)\). A quasigroup is said to be of type (1) if \({\mathcal M}(Q)\subseteq {\mathcal A}(Q)\) (the alternating group); of type (2) if \({\mathcal M}_{\ell}(Q)\subseteq {\mathcal A}(Q)\) and \({\mathcal R}(a,Q)\not\in {\mathcal A}(Q)\) for each \(a\in Q\); of type (3) if \({\mathcal M}_ r(Q)\subseteq {\mathcal A}(Q)\) and \({\mathcal L}(a,Q)\not\in {\mathcal A}(Q)\) for each \(a\in Q\); of type (4) if \({\mathcal L}(a,Q)\), \({\mathcal R}(a,Q)\not\in {\mathcal A}(Q)\) for each \(a\in Q\). The authors prove (among other statements about orthogonal automorphisms) the following theorems: If either \(n\geq 7\) is odd or n is divisible by 4, then there exists an orthomorphic idempotent quasigroup of type (1) and order n. If \(n\geq 5\) is odd, then there exists an orthomorphic idempotent quasigroup of type (2) (resp. (3)) and order n. If \(n\neq 15\) is odd, then there exists an orthomorphic idempotent quasigroup of type (4) and order n.
Reviewer: M.Csikós

MSC:

20N05 Loops, quasigroups
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
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