Non-tame automorphisms of extensions of periodic groups
If \(R\) is a characteristic subgroup of a free group \(F\), then an automorphism of \(F\) induces in a natural way an automorphism of \(F/R\). An automorphism of \(F/R\) induced in such a way is called tame. The question is whether the group \(F/R\) has non-tame automorphisms, i.e. automorphisms which cannot be induced by automorphisms of \(F\). The authors use the \(N\)-torsion invariant discovered by \textit{Y. Moriah} and \textit{M. Lustig} [J. Algebra 157, 170-198 (1993)], to show the existence of non-tame automorphisms of the following groups: (i) central extensions of free Burnside groups, i.e. the groups \(F_ n/[F^ p_ n,F_ n]\), where \(F_ n\) is a free group of finite rank \(n \geq 4\) and \(F^ p_ n\) is the subgroup generated by the \(p\)th powers of elements of \(F_ n\) for \(p\) a positive integer \(\geq 3\). (ii) The groups \(F_ n/\gamma_ c(F^ p_ n)\), where \(\gamma_ c(G)\) denotes as usual the \(c\)th term of the lower central series of the group \(G\). As a corollary they also prove that the group \(F_ n/[\gamma_ c(F_ n),F_ n]\) has non-tame automorphisms. In each case they produce specific automorphisms and they use the above mentioned invariant to prove that they are non- tame.
central extensions of free Burnside groups
scientific article; zbMATH DE number 474981
scientific article