On quasigroups rich in associative triples

Abstract

Let G be a group and G(*) a quasigroup on the ~.~ame underlying set. Let dist(G, G(*)) denote the number of pairs ( x , y ) e G 2 such that xy~x*y . For a finite quasig4"oup Q, n = card(Q), let t = dist(Q) = min dist(G, Q), where G ruu-~ through all groups with the same underlying set, and s = s(Q) the number of non-associative triples. Then 4 t n 2t 2 -24 t ~ s ~< 4m. If 1~,.~<3n2/32, then 3 m < s holds as well. Let n~>168 be an even integer and let t r = m i n s ( Q ) , where Q runs through all non-associative quasigroups of order n. Then t r= 16n 64.

DOI: 10.1016/0012-365X(83)90189-9

Cite this paper

@article{Drpal1983OnQR, title={On quasigroups rich in associative triples}, author={Ales Dr{\'a}pal}, journal={Discrete Mathematics}, year={1983}, volume={44}, pages={251-265} }