The structure of finite groups with weakly \(c\)-normal subgroups (Q722332)

Let \(G\) be a finite group. A subgroup \(X\) of \(G\) is said to be \textit{\(s\)-quasinormal} if \(XP=PX\) for each Sylow subgroup \(P\) of \(G\). This kind of subgroups was introduced and investigated by \textit{O. H. Kegel} [Math. Z. 78, 205--221 (1962; Zbl 0102.26802)] more than fifty years ago. Generalizing this concept, the subgroup \(X\) is called \textit{\(s\)-quasinormally embedded} if, for each prime divisor \(p\) of the order of \(X\), a Sylow \(p\)-subgroup of \(X\) is also a Sylow \(p\)-subgroup of some \(s\)-quasinormal subgroup of \(G\), see \textit{A. Ballester-Bolinches} and \textit{M. C. Pedraza-Aguilera} [J. Pure Appl. Algebra 127, No. 2, 113--118 (1998; Zbl 0928.20020)]. Recall also that the subgroup \(X\) is \textit{\(c\)-normal} if there exists a normal subgroup \(N\) of \(G\) such that \(G=XN\) and \(X\cap N\) is contained in the core \(X_G\) of \(X\) in \(G\). In the paper under review, the author studies a further embedding property: the subgroup \(X\) is said to be \textit{weakly \(c\)-normal} if there is a subnormal subgroup \(K\) of \(G\) such that \(G=XK\) and \(X\cap K\) is \(s\)-quasinormally embedded in \(G\). The influence of the existence of weakly \(c\)-normal subgroups on the structure of a finite group is studied.