On Frattini subloops and normalizers of commutative Moufang loops

TL;DR

This paper investigates the structure of commutative Moufang loops (CMLs), focusing on their Frattini subloops and normalizers, establishing conditions for when the Frattini subloop equals the loop itself, and analyzing the structure of their multiplication groups.

Contribution

It characterizes the Frattini subloop in CMLs, introduces the notion of normalizers for subloops, and describes the structure of the multiplication group in relation to these properties.

Findings

$rak F(L) = L$ iff $rak F(rak M) = rak M$

CMLs satisfy the normalizer condition when $rak F(L) eq L$

Divisible subgroups of $rak M$ are abelian and serve as direct factors

Abstract

Let $L$ be a commutative Moufang loop (CML) with multiplication group $\frak M$, and let $\frak F(L)$, $\frak F(\frak M)$ be the Frattini subgroup and Frattini subgroup of $L$ and $\frak M$ respectively. It is proved that $\frak F(L) = L$ if and only if $\frak F(\frak M) = \frak M$ 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 $\frak F(L) \neq L$ then $L$ satisfies the normalizer condition and that any divisible subgroup of $\frak M$ is an abelian group and serves as a direct factor for $\frak M$.