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IMCS/Publications/BASM/Issues/BASM n.3 (70), 2012/

On Frattini subloops and normalizers of commutative Moufang loop

Authors: N. I. Sandu
Keywords: Commutative Moufang loop, multiplication group, Frattini subloop, Frattini subgroup, normalizer, loop with normalizer condition, divisible loop.

Abstract

Let $L$ be a commutative Moufang loop (CML) with the multiplication group $\frak M$, and let $\frak F(L)$, $\frak F(\frak M)$ be the Frattini subloop of $L$ and Frattini subgroup of $\frak M$. It is proved that $\frak F(L) = L$ if and only if $\frak F(\frak M) = \frak M$, 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 $\frak F(L) \neq L$, 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 $\frak M$.

N. I. Sandu
Tiraspol State University
str. Iablochkin, 5, Chisinau, MD-2069
Moldova
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