AUTOMORPHISMS OF CERTAIN RELATIVELY FREE GROUPS AND LIE ALGEBRAS

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.

THE 2-GENERATOR RESTRICTED BURNSIDE GROUP OF EXPONENT 7

We report on our construction of a power-commutator presentation for R(2,7), the largest finite 2-generator group of exponent 7. Our calculations show that R(2,7) has order 720416, nilpotency class 28, and derived length 5. The calculations also imply that the associated Lie ring of R(2,7) satisfies relations which are not consequences of the multilinear identities which hold in the associated Lie rings of groups of exponent 7.

On the Associated Lie Ring and the Adjoint Group of a Radical Ring

We investigate connections between the associated Lie ring and the adjoint group of a radical ring, studying their upper central chains. Part of a conjecture of S. A. Jennings is proved, and one of our results improves a theorem of his.