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Every Moufang loop of odd order with nontrivial commutant has nontrivial center

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Abstract

We get a partial result for Phillips’ problem: does there exist a Moufang loop of odd order with trivial nucleus? First we show that a Moufang loop Q of odd order with nontrivial commutant has nontrivial nucleus, then, by using this result, we prove that the existence of a nontrivial commutant implies the existence of a nontrivial center in Q. Introducing the notion of commutantly nilpotence, we get that the commutantly nilpotence is equivalent to the centrally nilpotence for the Moufang loops of odd order.

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Authors and Affiliations

  1. Department of Algebra and Number Theory, Eötvös University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary

    Piroska Csörgő

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  1. Piroska Csörgő
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Correspondence to Piroska Csörgő.

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This paper was partially supported by Hungarian National Foundation for Scientific Research Grant # K84233.

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Csörgő, P. Every Moufang loop of odd order with nontrivial commutant has nontrivial center. Arch. Math. 100, 507–519 (2013). https://doi.org/10.1007/s00013-013-0526-z

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  • DOI: https://doi.org/10.1007/s00013-013-0526-z

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