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- W2050862419 abstract "Let E be a given finitely generated Commutative Moufang Loop (CML for short); let k ( E ) be its associative central nilpotency class, Φ ( E ) its Frattini subloop and Γ ( E ) the smallest normal subloop of E such that k ( E / Γ ( E ) ) < k ( E ). Using mainly the canonical mappings from G = Aut( E ) into Aut( E /Φ( E ) ) or Aut( E /r(E)) we shall give upper bounds for the order of G . More accurate results may be stated if E is assumed to have exponent 3, or when E is free in some sense. If E is the free loop on n generators among the CMLs (resp. exponent 3 CMLs) whose nilpotency class is at most 2, then the exact sequence 0 → K e r f → G → f A u t ( E / Γ ( Ε ) ) → 0 splits so that an explicit description of G may be given: it is some semi-direct product of a linear group by an elementary abelian 3-group of dimension n ( 3 n ) . Particularly the well-known loop L (3) , which is the free exponent 3 CML on 3 generators, has a semi-direct product of G L ( 3 , F 3 ) by F 3 3 for its group of automorphisms. In another connection, ∣Aut( L (4) ∣ = 2 9 .3 38 .5.13 where L (4) is the free exponent 3 CML on 4 generators." @default.
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- W2050862419 date "1980-12-01" @default.
- W2050862419 modified "2023-10-03" @default.
- W2050862419 title "Groupes d'Automorphismes des Boucles de Moufang Commutatives" @default.
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- W2050862419 doi "https://doi.org/10.1016/s0195-6698(80)80029-1" @default.
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