\(m\)-Wielandt series in infinite groups (Q2726651)

Let \(G\) be a group, and let \(m\) be a positive integer. The intersection of the normalizers of all subnormal subgroups of \(G\) with defect at most \(m\) will be denoted by \(u_m(G)\), and the \(m\)-Wielandt series of \(G\) is defined inductively by the positions \(u_{m,0}(G)=\{1\}\) and \(u_{m,i}(G)/u_{m,i-1}(G)=u_m(G/u_{m,i-1}(G))\). The group \(G\) is said to be a \({\mathcal W}_m^*\)-group (or to have finite \(m\)-Wielandt length) if \(u_{m,n}(G)=G\) for some non-negative integer \(n\); moreover, \(G\) is a \({\mathcal W}^*\)-group if it has finite \(m\)-Wielandt length for each positive integer \(m\). It is proved that all nilpotent-by-finite groups have the property \({\mathcal W}^*\), and that an \({\mathcal S}_1\)-group belongs to the class \({\mathcal W}^*\) if and only if it is (nilpotent divisible)-by-(nilpotent torsion-free)-by-finite (here a group \(G\) is said to be an \({\mathcal S}_1\)-group if it is a soluble group with finite Abelian section rank whose elements have only finitely many distinct prime orders).