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Octonions, simple Moufang loops and triality. (English) Zbl 1062.20069

The authors deal with the classification of non-associative simple Moufang loops. It is known [M. W. Liebeck, Math. Proc. Camb. Philos. Soc. 102, 33-47 (1987; Zbl 0622.20061)] that there exists only one class of nonassociative finite simple Moufang loops, the so called Paige loops, which can be constructed over every finite field. The authors present a new proof of this result by establishing a correspondence between (simple) Moufang loops, (simple) Moufang 3-nets and (\(S\)-simple) groups with triality \((G,S)\).
Since the structure of the latter groups is rather transparent, they can be easily classified (cf. Liebeck), and this yields the result on simple Moufang loops as a corollary. The paper further investigates the automorphism groups of Paige loops over perfect fields, generators of finite Paige loops and integral Cayley numbers. The discussion of the core results is preceded by an ample survey on loops, 3-nets and composition algebras, which makes the paper interesting and accessible to a broader audience.

MSC:

20N05 Loops, quasigroups
20D05 Finite simple groups and their classification
17A75 Composition algebras
53A60 Differential geometry of webs

Citations:

Zbl 0622.20061
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