Automorphism groups of free metabelian nilpotent groups (Q1191181)

Let \(\text{Aut}(M_{n,c})\), \(n \geq 2\), \(c \geq 2\), be the automorphism group of the free \(n\)-generator metabelian nilpotent group \(M_{n,c}\) of class \(c\) with generators \(\{x_ 1, \dots,x_ n\}\) and let \(T \leq \text{Aut} (M_{n,c})\) be the subgroup consisting of all tame automorphisms of \(M_{n,c}\). If \(c \leq 2\) then \(T = \text{Aut}(M_{n,c})\) whereas for \(c \geq 3\), \(\text{Aut} (M_{n,c})\) does contain non-tame automorphisms. Here the authors are interested in the problem of finding an economical generating set for \(\text{Aut}(M_{n,c})\) in the general case \(n \geq 2\), \(c \geq 3\). Define \(\theta_ k\), \(3 \leq k \leq c\), by \(x_ 1 \to x_ 1[x_ 1, x_ 2, (k-2)x_ 1]\), \(x_ i \to x_ i\), \(i \neq 1\), and set \(K = \langle T,\theta_ 3, \dots, \theta_ c \rangle\). In this note is is proved that some power of every automorphism of \(M_{n,c}\) lies in \(K\).