Automorphism groups of free nilpotent groups
Let \(F_{n,c}\) be the free nilpotent group of class c and rank n, where c and n are positive integers. \textit{S. Andreadakis} [Arch. Math. 42, 296- 300 (1984; Zbl 0522.20025)] proved that for \(n\geq c\geq 2\) the automorphism group of \(F_{n,c}\) is generated by tame automorphisms (i.e. automorphisms induced by automorphisms of the absolutely free group of rank n) together with certain automorphisms \(\theta _ 3,...,\theta _ c\). In the present paper a simpler result is obtained: for \(n\geq 2\) and \(n\geq c-1\) the automorphism group of \(F_{n,c}\) is generated by tame automorphisms together with the automorphism \(\delta\) satisfying \(x_ 1\delta =x_ 1[x_ 1,x_ 2,x_ 1]\) and \(x_ i\delta =x_ i\) for all \(i>1\), where \(x_ 1,...,x_ n\) are free generators of \(F_{n,c}\). Thus for \(n\geq 4\) and \(n\geq c-1\) the automorphism group of \(F_{n,c}\) is generated by three elements. A more complicated result is also obtained in the case where \(n\geq 2\) and \((c+1)\leq n\leq c-1\).
scientific article; zbMATH DE number 4040172