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 (g)P = P(g) 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).Generalizing the notion of the center, Ore [10] defined the quasicenter Q{G) of a group G to be the subgroup generated by all elements g of G such that (g}H = H(g} for all H < G. Mukherjee [9] determined its structure.He also defined and studied the hyperquasicenter Q*(G) of G which generalizes the concept of the hypercenter.Here we define and investigate another center and hypercenter of a finite group.We say, following Kegel [7], that a subgroup of G is vr-quasinormal in G if it permutes with every Sylow subgroup of G and define the generalized center ZGn(G) of G to be the subgroup generated by all elements g of G such that (g) is 77-quasinormal in G.This leads to the definition of the generalized hypercenter ZGn(G).Let {ZGn{G ))0 = 1 and (ZGn(G )), + 1/(ZGn(G)),-be the generalized center of G/(ZGn(G)).This yields an ascending chain of characteristic subgroupsThe terminal member ZGn(G) is called the generalized hypercenter.The main results are: (1) If <g> is vr-quasinormal in G, so is every subgroup(4) ZGn(G) is supersolvable; (5) ZGn(G) is contained in the intersection of the maximal supersolvable subgroups of G; and (6) ZGn(G) is the product of all generalized hypercentral subgroups of G.