Some results on the generalized hypercenter of finite groups. (Q558284)

A subgroup of a group \(G\) is called quasinormal if it permutes with all subgroups of \(G\) and it is called \(S\)-quasinormal if it permutes with all Sylow subgroups of \(G\). The generalized center \(\text{genz}(G)\) of \(G\) is the group generated by all cyclic, \(S\)-quasinormal subgroups of \(G\). Finally the generalized hypercenter \(\text{genz}_\infty(G)\) is the final term of the series \(1=\text{genz}_0(G)\leq\text{genz}_1(G)\leq\text{genz}_2(G)\leq\cdots\) where \(\text{genz}_i(G)/\text{genz}_{i-1}(G)=\text{genz}(G/\text{genz}_{i-1}(G))\). The authors prove various results about the structure of a finite group \(G\) (like \(p\)-nilpotency, supersolvability, etc.) under the assumption that the generalized hypercenter contains certain subgroups of \(G\). This generalizes previous work of \textit{M. Assad} and \textit{M. Ezzat Mohamed} [Commun. Algebra 29, No. 5, 2239-2244 (2001; Zbl 0991.20018)].