New characterizations of \(p\)-nilpotency and Sylow tower groups. (Q360978)

Introduced by \textit{M. C. Pedraza-Aguilera} and the reviewer [J. Pure Appl. Algebra 127, No. 2, 113-118 (1998; Zbl 0928.20020)] and growing out of the concepts of Sylow permutability and permutable embedding, a subgroup \(H\) of a finite group \(G\) is said to be S-permutably embedded in \(G\) if a Sylow \(p\)-subgroup of \(G\) is a Sylow \(p\)-subgroup of some S-permutable subgroup of \(G\). In the paper under review, the author extends this definition and defines a subgroup \(H\) of a finite group \(G\) to be \(s^*\)-permutably embedded in \(G\) if there exists a subgroup \(T\) of \(G\) such that \(G=HT\) and \(H\cap T\) is contained in an S-permutably embedded subgroup contained in \(H\). This subgroup embedding property generalizes many other ones already appearing in the literature such as c-nomality, c-supplementation, weakly S-permutability and weakly S-permutable embedding, and it is used in the paper to generalise many known results which establish conditions for when a finite group is \(p\)-nilpotent or belongs to a saturated formation containing Sylow tower groups. Reviewer's remark: It would have been helpful for the reader to have the explicit definition of S-permutable embedding in the paper.