On commutative Moufang loops with some restrictions for subloops and subgroups of its multiplication groups

Natalia Lupashco

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2009), pp. 52-56

Résumé

It is proved that if an infinite commutative Moufang loop $L$ has such an infinite subloop $H$ that in $L$ every associative subloop which has with $H$ an infinite intersection is a normal subloop then the loop $L$ is associative. It is also proved that if the multiplication group $\mathfrak M$ of infinite commutative Moufang loop $L$ has such an infinite subgroup $\mathfrak N$ that in $\mathfrak M$ every abelian subgroup which has with $\mathfrak N$ an infinite intersection is a normal subgroup then the loop $L$ is associative.