Two notes on free soluble groups

A. L. Šhmelkin

Algebra i logika, Tome 6 (1967) no. 2, pp. 95-109.

Résumé

Let $G$ be the free soluble group of length $k>0$ ($G^{(k)}=1$), and let $\varphi$ be an automorphism of $G$. If $x\varphi=x$ for all $x\in G^{(k-1)}$, then $\varphi$ is the inner automorphism induced by an element $y\in G^{(k-1)}$. We study also a question: is it true that in the free soluble group $G$ $\{y\}^G=\{h\}^G$ if and only if $y^{\pm1}=c^{-1}hc$? (Fоr absolutely free group this is a theorem of W. Magnus). In particular cases But this is not true in general; two elements $g, h$ of the free metabelian group $G$ of the rank $2$ are constructed in the paper, such that $\{g\}^G=\{h\}^G$ and $g^{\pm1}, h$ are not conjugated.