WEAKLY HYPERCENTRAL SUBGROUPS OF FINITE GROUPS

In this article the study of generalized Frattini subgroups of finite groups, developed byJ. C. Beidleman and T. K. Seo, is continued. We call a proper normal subgroup H of a finite group G, a special generalized Frattini subgroup of G provided that G = NG(A) for each normal subgroup L of G and each Hall subgroup A of L such that G = HNG(A). Z. Janko proved that a subnormal subgroup K of a finite group G is Π-closed, Π is a set of primes, whenever K/(K∩Ø(G)) is Π-closed, where Ø(G) denotes the Frattini subgroup of G. We prove that a subnormal subgroup K of a finite group G is Π-closed whenever K/(K∩H) is Π-closed where H is a special generalized Frattini subgroup of G. From this result we prove that a proper normal subgroup H of a finite group G is a special generalized Frattini subgroup of G if and only if if is a weakly hypercentral subgroup of G. © 1969 by Pacific Journal of Mathematics.