On Frattini subloops and normalizers of commutative Moufang loops

N. I. Sandu

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2012), pp. 16-27

Résumé

Let $L$ be a commutative Moufang loop (CML) with the multiplication group $\mathfrak M$, and let $\mathfrak F(L)$, $\mathfrak F(\mathfrak M)$ be the Frattini subloop of $L$ and Frattini subgroup of $\mathfrak M$. It is proved that $\mathfrak F(L)=L$ if and only if $\mathfrak F(\mathfrak M)=\mathfrak M$, and the structure of this CML is described. The notion of normalizer for subloops in CML is defined constructively. Using this it is proved that if $\mathfrak F(L)\neq L$, then $L$ satisfies the normalizer condition and that any divisible subgroup of $\mathfrak M$ is an abelian group and serves as a direct factor for $\mathfrak M$.