On Frattini subloops and normalizers of commutative Moufang loops

Abstract

Let $L$ be a commutative Moufang loop (CML) with multiplication group $\mathfrak{M}$, and let $\mathfrak{F}(L)$, $\mathfrak{F}(\mathfrak{M})$ be the Frattini subgroup and Frattini subgroup of $L$ and $\mathfrak{M}$ respectively. It is proved that $\mathfrak{F}(L)=L$ if and only if $\mathfrak{F}(\mathfrak{M})=\mathfrak{M}$ and is described the structure of this CLM. Constructively it is defined the notion of normalizer for subloops in CML. 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 $\mathfrak{M}$.

15 pages

References in corpus (1)

• The commutative Moufang loops with minimum conditions for subloops II