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Orthogonality of transitive, distributive quasi-groups corresponding to finite cyclic groups. (Russian) Zbl 0635.20038
The paper deals with transitive, distributive quasigroups (TDQ) corresponding to finite abelian groups and proves, if the order of a cyclic group is a) an even number, then a TDQ does not exist, b) an odd number, then there exists at least one TDQ. Moreover, if the order of a finite abelian group G is \(n=p_ 1^{k_ 1}p_ 2^{k_ 2}...p_ t^{k_ t}\) or \(n=2^ kp_ 1^{k_ 1}...p_ s^{k_ s}\), where \(k_ i>0\), \(k\geq 2\), \(p_ i\neq 2\) are prime numbers and \(G=G_ 1\dot +G_ 2\dot +...\dot +G_ k\dot +G_{k+1}\dot +...\dot +G_ s\), \(| G_ j| =2\), \(1\leq j\leq k\), \(| G_ h| =p_ h^{k_ h}\), \(k+1\leq h\leq s\), then the corresponding TDQ exists. In the case that there exist several TDQ corresponding to a given finite abelian group, then it is determined a necessary and sufficient condition when two TDQs are orthogonal.
Reviewer: E.Brozikov√°
20N05 Loops, quasigroups
20K01 Finite abelian groups
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