Perfect radicals and homology of group extensions
If G is a group, \({\mathcal P}G\) denotes the perfect radical of G. An epimorphism \(\phi\) : \(G\to Q\) with \(\phi\) (\({\mathcal P}G)={\mathcal P}Q\) is called an epimorphism preserving perfect radicals \((EP^ 2R)\). The motivation for study is topological, namely certain questions about the plus construction. It is easy to see that \(\phi\) is \(EP^ 2R\) (with \(N=Ker \phi)\) if one of the following holds: (i) G splits over N, (ii) \(G^{(m)}\leq N({\mathcal P}G)\) for some finite m, (iii) \(N^{(m)}\leq {\mathcal P}G\) for some finite m. However the property fails if G is free and Q is perfect. The authors give a number of examples disproving conjectures about \(EP^ 2R\) (and thereby certain topological conjectures). For example, they construct a group G with \(N\triangleleft G\) such that \({\mathcal P}G=1\), \(Q=G/N\) is perfect and non-trivial and elements of G induce inner automorphisms in N. (Thus Q operates trivially on the integral homology of N.) The group G is an interdirect product of nilpotent groups.
scientific article; zbMATH DE number 4002371