AUTOMORPHISMS OF CERTAIN RELATIVELY FREE GROUPS AND LIE ALGEBRAS
Abstract
For positive integers n and c, with n≥2, let Gn,c be a relatively free group of rank n in the variety N2A∧AN2∧Nc. It is shown that there exists an explicitly described finite subset Ω of IA-automorphisms of Gn,c such that the cardinality of Ω is independent upon n and c and the subgroup of the automorphism group Aut(Gn,c) of Gn,c generated by the tame automorphisms and Ω has finite index in Aut(Gn,c). This is a simpler result than one given in [12, Theorem 1(I)]. Let L(Gn,c) be the associated Lie ring of Gn,c and K be a field of characteristic zero. The method developed in the proof of the aforementioned result is applied in order to find an explicitly described finite subset ΩL of the IA-automorphism group of K⊗L(Gn,c) such that the automorphism group of K⊗L(Gn,c) is generated by GL(n,K) and ΩL. In particular, for n≥3, the cardinality of ΩL is independent upon n and c.
