Quelques consequences de la locale nilpotence des boucles de moufang commutatives
作者:
Lucien Bénéteau,
期刊:
Communications in Algebra
(Taylor Available online 1983)
卷期:
Volume 11,
issue 15
页码: 1725-1753
ISSN:0092-7872
年代: 1983
DOI:10.1080/00927878308822929
出版商: Gordon and Breach Science Publishers Ltd.
数据来源: Taylor
摘要:
It is well-known that any finitely generated commutative Moufang loop (CML) is centrally nilpotent and has a finite derived subloop. Consequently such a loop possesses all the classical properties of noethe-rianity: any subloop is finitely generated too, any surjective endomorphism is an automorphism, etc. Besides we prove that, in any CML E(finitely generated or not) the maximal subloops are normal of prime index ; thus the Frattini quotient E/Φ(E)is an abelian group, sub-direct product of groups of prime order. We shall study also some dual notion of the Frattini subloop, namely the subloop φ*(E) generated by the minimal normal subloops ; it turns out that φ*(E) is made up by the products of the prime order central elements.
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