Une conjecture sur les suites centrales d’une boucle de Moufang commutative libre


The lower and upper central series {C<sup>i</sup>(E)} and {Z<sub>j</sub>(E)} of a CML (commutative Moufang loop) E are defined just as the central series of a group, the associators (x,y,z)=(x(yz))^[-1} ((xy)z) playing the same role as the commutators for groups.As was skown recently, if E=𝕃<sub>n</sub> (resp.L<sub>n</sub>) is the free CML (resp. exponent 3 CML) on n≥ 3 generators, the common length of the central series is exactly n-1. Besides Z<sub>1</sub>(L<sub>n</sub>) contains a torsion-free abelian group A<sub>n</sub> of rank n such that 𝕃<sub>n</sub>=L<sub>n</sub>/A<sub>n</sub>.In view of WITT's result about the central series of "the free nilpotent groups of bounded class" we conjecture that the inclusion: C<sup>i</sup>⊂ Z_{n-1-i} is in fact an equality in L<sub>n</sub>.In 𝕃<sub>n</sub>, this would imply that Z_{n-1-i} is the direct product of C<sup>i</sup> by A<sup>n</sup>.The required equalities will be actually checked when either i=1 or $n≤ 4.

DOI Code: 10.1285/i15900932v3n1p45

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