Quasigroups satisfying balanced but not Belousov equations are group isotopes (Q1180545)

The authors discuss the problem what quasigroups ( loops) are isotopic to a group. They prove the theorem saying that a quasigroup \((Q,*)\) is isotopic to a group if and only if the Reidemeister condition \((x_ 1*y_ 2=x_ 2*y_ 1\), \(x_ 1*y_ 4=x_ 2*y_ 3\), \(x_ 4*y_ 1=x_ 3*y_ 2\) implies \(x_ 3*y_ 4=x_ 4*y_ 3)\) holds. From that it follows the theorem stated in the title.