Vol. 31, No. 2, 1969

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Weakly hypercentral subgroups of finite groups

Donald Campbell Dykes

Vol. 31 (1969), No. 2, 337–346
Abstract

In this article the study of generalized Frattini subgroups of finite groups, developed by J. 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 = N𝜃(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 ϕ(G)) is π-closed, where ϕ(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 H is a weakly hypercentral subgroup of G.

Mathematical Subject Classification
Primary: 20.54
Milestones
Received: 16 September 1968
Published: 1 November 1969
Authors
Donald Campbell Dykes