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Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, 2012, номер 3, страницы 16–27
(Mi basm322)
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On Frattini subloops and normalizers of commutative Moufang loops
N. I. Sandu Tiraspol State University, str. Iablochkin, 5, Chisinau, MD-2069, Moldova
Аннотация:
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 $\mathfrak M$.
Ключевые слова и фразы:
commutative Moufang loop, multiplication group, Frattini subloop, Frattini subgroup, normalizer, loop with normalizer condition, divisible loop.
Поступила в редакцию: 06.04.2011
Образец цитирования:
N. I. Sandu, “On Frattini subloops and normalizers of commutative Moufang loops”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2012, no. 3, 16–27
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/basm322 https://www.mathnet.ru/rus/basm/y2012/i3/p16
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