The linearity of automorphism groups of relatively free groups
Let \({\mathcal N}_c\) be the variety of all nilpotent groups of class \(c\) (\(c\geq 1\)), \(\mathcal U\) be the variety of all abelian groups, \({\mathcal U}_k\) be the variety of all abelian groups of exponent \(k\), \({\mathcal B}_m\) be the variety of all locally finite groups of exponent \(m\) and \(\mathcal I\) be the variety of all groups. The following main result is obtained. Theorem. Let \(F_n({\mathcal M})\) be a free group of rank \(n\) of a variety \({\mathcal M}\neq{\mathcal I}\) and \(n\geq 2\). 1. If \({\mathcal M}\nsubseteq{\mathcal N}_c{\mathcal U}{\mathcal B}_m\), then \(\text{Aut }F_n({\mathcal M})\) is not linear. 2. Assume that \({\mathcal M}\subseteq{\mathcal N}_c{\mathcal U}{\mathcal B}_m\) and \({\mathcal U}_k{\mathcal U}\nsubseteq{\mathcal M}\). Then the group \(\text{Aut }F_n({\mathcal M})\) is linear. 3. Assume that \({\mathcal M}\subseteq{\mathcal N}_c{\mathcal U}{\mathcal B}_m\) and \({\mathcal U}_k{\mathcal U}\subseteq{\mathcal M}\). Then \(\text{Aut }F_n({\mathcal M})\) is not linear.
scientific article; zbMATH DE number 898951