Une conjecture sur les suites centrales d’une boucle de Moufang commutative libre
Abstract
The lower and upper central series
and
of a CML (commutative Moufang loop) E are defined just as the central series of a group, the associators
playing the same role as the commutators for groups.As was skown recently, if
(resp.
) is the free CML (resp. exponent 3 CML) on
generators, the common length of the central series is exactly
. Besides
contains a torsion-free abelian group
of rank n such that
.In view of WITT's result about the central series of "the free nilpotent groups of bounded class" we conjecture that the inclusion:
is in fact an equality in
.In
, this would imply that
is the direct product of
by
.The required equalities will be actually checked when either i=1 or $n≤ 4.
![{C<sup>i</sup>(E)}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/6bedd6bf73a28bd59d8b367b11f44bf8.png)
![{Z<sub>j</sub>(E)}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/153b97519da1693dd2ed5eeaa04d8529.png)
![(x,y,z)=(x(yz))^[-1} ((xy)z)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/b6553d9abf6d396a079a28d6571f849b.png)
![E=𝕃<sub>n</sub>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8ff020d6af4e4ed32f989544ea3c5ea4.png)
![L<sub>n</sub>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/52fbf8c5d8b1b7ed77a8bb379a9f1b41.png)
![n≥ 3](http://siba-ese.unile.it/plugins/generic/latexRender/cache/a897948527da6a066bdd1faab3fa0651.png)
![n-1](http://siba-ese.unile.it/plugins/generic/latexRender/cache/a438673491daae8148eae77373b6a467.png)
![Z<sub>1</sub>(L<sub>n</sub>)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/23d53d9b64e40932e4456be4b5d2a6a9.png)
![A<sub>n</sub>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/932865e643aacccf2c4b0f9bc770b0c0.png)
![𝕃<sub>n</sub>=L<sub>n</sub>/A<sub>n</sub>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/cde34e72f79de2d48ac9b827dfda7c2a.png)
![C<sup>i</sup>⊂ Z_{n-1-i}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/1ef4a538cc7de03bac24b4218442251c.png)
![L<sub>n</sub>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/52fbf8c5d8b1b7ed77a8bb379a9f1b41.png)
![𝕃<sub>n</sub>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/128f79b0a19514ab4d1668947ab636e3.png)
![Z_{n-1-i}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/b9061cf342703f366aa743a93751f74a.png)
![C<sup>i</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/7ffaee4e4ef84eb902f8bcdf836f18df.png)
![A<sup>n</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/c05ae58d60be36795cb3193a0a2f92c6.png)
DOI Code:
10.1285/i15900932v3n1p45
Full Text: PDF