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 $\mathfrak{F}(L)\ne L$, 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 $\mathfrak{M}$.

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