Unipotent automorphisms of solvable groups (Q524635)

An automorphism \(\alpha\) of a group \(G\) is called a \textit{nil-automorphism} if for each element \(g\) of \(G\) there exists a positive integer \(n=n(g)\) such that \([g,_n\alpha]=1\). If the integer \(n\) can be chosen independently on \(g\), \(\alpha\) is said to be \textit{\(n\)-unipotent} (or simply unipotent). Clearly, nil-automorphisms are strictly related to Engel elements of groups, and hence it is a natural question to ask under which conditions a group consisting of unipotent automorphisms needs to be locally nilpotent. The main result of the paper under review states that if \(G\) is a soluble group and \(\Gamma\) is a finitely generated soluble subgroup of \(\Aut(G)\) consisting of \(n\)-unipotent automorphisms, then \(\Gamma\) stabilizes a finite series of \(G\), and so it is nilpotent. It is easy to see that there exists a soluble group \(G\) admitting an abelian group \(\Gamma\) consisting of \(3\)-unipotent automorphisms such that \(\Gamma\) does not stabilize any finite series of \(G\). Moreover, the authors construct a group \(G\) for which there exists an automorphism \(\alpha\) which is \(n\)-unipotent for some \(n\) and does not stabilize any finite series of \(G\).