Subnormality and residuals for saturated formations: a generalization of Schenkman's theorem (Q2037530)

In this paper, only finite groups are considered by the authors. Let \(G\) be a finite group. The authors prove the following generalization of Schenkman's theorem on the centraliser of the nilpotent residual of a subnormal subgroup: Let \(\mathcal {F}\) be a hereditary saturated formation containing all nilpotent groups. Suppose that \(S\) is a generalized subnormal subgroup of \(G\), then the centraliser of the \(\mathcal {F}\)-residual of \(S\) is self-centralizing. This improves the result by \textit{E. Schenkman} [Pac. J. Math. 5, 995--998 (1955; Zbl 0067.25902)].