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Moufang loops of class 2 and cubic forms. (English) Zbl 0962.20046

The author shows that: – the nucleary-derived (normal associator) subloop of a Moufang loop of class \(p\) has exponent dividing 6. In particular, for \(p>3\) the subloop of elements of \(p\)-power order is associative; – if \(L\) is an SFML (small Frattini Moufang loop), \(Z\) its central subgroup of order \(p\), then \(L/Z\) has the structure of a vector space with a symplectic cubic form; – every symplectic cubic form is realized by some SFML; – two SFMLs are isomorphic iff their symplectic cubic spaces are isomorphic. Finally, many of these results are extended to all finite Moufang loops of class 2.

MSC:

20N05 Loops, quasigroups
11E76 Forms of degree higher than two
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