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

Let $\mathfrak M$ be the multiplication group of a commutative Moufang loop $Q$. In this paper it is proved that if all infinite abelian subgroups of $\mathfrak M$ are normal in $\mathfrak M$, then $Q$ is associative. If all infinite nonabelian subgroups of $\mathfrak M$ are normal in $\mathfrak M$, then all nonassociative subloops of $Q$ are normal in $Q$, all nonabelian subgroups of $\frak M$ are normal in $\mathfrak M$ and the commutator subgroup $\mathfrak M'$ is a finite 3-group.