Generalized center and hypercenter of a finite group

Abstract:

The generalized center of a group $G$ is defined to be the subgroup generated by all elements $g$ of $G$ such that $\left \langle g \right \rangle P = P\left \langle g \right \rangle$ for all Sylow subgroups $P$ of $G$. This generalizes the concept of the center and quasicenter and leads to the notion of the generalized hypercenter which is defined in the same way as the hypercenter and hyperquasicenter. It is shown that the generalized center is nilpotent and the generalized hypercenter is supersolvable (in fact, the generalized hypercenter is contained in the intersection of the maximal supersolvable subgroups).