On the hyperquasicenter of a group (Q2711631)

An element \(x\) of a group \(G\) is quasicentral if \(H=\langle x\rangle\) is a permutable subgroup of \(G\), i.e. \(HX=XH\) for all subgroups \(X\) of \(G\). The hyperquasicenter \(Q^*(G)\) is the subgroup generated by all quasicentral elements of \(G\). Corresponding to the hypercenter the hyperquasicenter \(Q^*(G)\) of a group \(G\) is defined as the largest term of the upper quasicentral chain of normal subgroups of \(G\). It is for instance shown that the hyperquasicenter of a group \(G\) is the largest hypercyclically embedded normal subgroup \(\sigma(G)\) of \(G\), i.e. \(\sigma(G)\) has an ascending \(G\)-invariant series whose factors are cyclic and the factor group \(G/\sigma(G)\) has no nontrivial cyclic normal subgroups. Further properties of \(\sigma(G)\) and \(Q^*(G)\) are investigated.