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.
