Normalizers of nilpotent residuals
Let \(G\) be a finite group. \textit{L. Gong} et al. [J. Algebra Appl. 15, No. 1, Article ID 1650008, 9 p. (2016; Zbl 1342.20017)] have noted that if a subgroup \(M\) of \(G\) normalises the derived subgroup of every non-subnormal subgroup of \(G\), then \(M\) does not necessarily normalises the derived subgroup of every subgroup. On the other hand, the authors prove the following. If a subgroup \(M\) of \(G\) normalises the nilpotent (soluble) residual of every non-subnormal subgroup of \(G\), then \(M\) normalises the nilpotent (soluble) residual of every subgroup. The authors ask whether the analogous result holds for supersolvable residuals, this question has been answered affirmatively by \textit{A. Ballester-Bolinches} et al. [Arch. Math. 109, No. 4, 305--310 (2017; Zbl 1378.20017)].
scientific article; zbMATH DE number 6688680