Talk:Moufang loop

From Wikipedia, the free encyclopedia

(1) Re - M(S3,2)is the smallest nonassociative Moufang loop, which has order 12. Can anyone provide a Cayley table for this? I can only find the three associative groups, and a query to sci.math (etc) elicited no response.

(2) Vector division and conservation. In Mathsource/4894 & /6189 I (empirically) establish that if Moufang Loops are used as multiplication tables for vectors, every vector has a multiplicative inverse, and the factors of the determinant of the symbolic loop table are conserved on multiplication and division; they are denominators for a partial fraction formulation of this inverse. This leads to the formulation of many named algebras. Would a note on this be acceptable? I ask on the discussion page because an entry on ArcTanh was rejected as "original research". Roger Beresford 08:14, 20 May 2006 (UTC)

You should be able to reconstruct the Cayley table for M(S_3,2) given the Cayley table for S_3 and the information given in the article. Regarding point 2: it sounds like what you are describing constitutes original research. See the offcial policy if you are in doubt. -- Fropuff 04:01, 22 May 2006 (UTC)

Thanks. (1)Unfortunately my efforts only give an associative table. (2) True. The rest is silence. Roger Beresford 14:29, 22 May 2006 (UTC)

I can at least explain why M(S₃,2) is nonassociative. Let g and h be two noncommuting elements of S₃ (such as (1 2 3) and (1 2)). Then

g(hu) = (hg)u ≠ (gh)u.

So g, h and u form a nonassociative triple. -- Fropuff 20:18, 22 May 2006 (UTC)

If you really want a Cayley table, download the LOOPS package for GAP mentioned in the references. Alternatively, see the reference to Goodaire et al. --Michael Kinyon 15:14, 3 August 2006 (UTC)