Minimally nonassociative commutative Moufang loops (Q5937362)

Let \(L\) be a loop with identity 1. Then the associator \((x,y,z)\) of three elements \(x,y,z\) in \(L\) is defined by the equation \((xy)z=(x(yz))(x,y,z)\). The authors prove the theorem: ``Let \(L\) be a commutative Moufang loop which is generated by three elements \(a\), \(b\) and \(c\) which do not associate, and let \(x,y,z\) be any three elements of \(L\). Then \((x,y,z)\neq 1\) if and only if \((x,y,z)=(a,b,c)\) (\(=L\))''. Corollary: A commutative Moufang loop \(L\) which is not associative has the property that all proper subloops are associative if and only if \(L\) can be generated by three elements.