Groupes d'Automorphismes des Boucles de Moufang Commutatives

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→Kerf→G→fAut(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(3n). Particularly the well-known loop L(3), which is the free exponent 3 CML on 3 generators, has a semi-direct product of GL(3,F3) by F33 for its group of automorphisms. In another connection, ∣Aut(L(4)∣ = 29.338.5.13 where L(4) is the free exponent 3 CML on 4 generators