## 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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