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The purpose of this paper is to
generalize some of the fundamental properties of the Frattini subgroup of a finite
group. For this purpose we call a proper normal subgroup H of G a generalized
Frattini subgroup if and only if G = NG(P) for each normal subgroup L of G and
each Sylow p-subgroup P,p is a prime, of L such that G = HNG(P). Here NG(P) is
the normalizer of P in G. Among the generalized Frattini subgroups of a finite
nonnilpotent group G are the center, the Frattini subgroup, and the intersection
L(G) of all selfnormalizing maximal subgroups of G. The product of two
generalized Frattini subgroups of a group G need not be a generalized Frattini
subgroup, hence G may not have a unique maximal generalized Frattini
subgroup.
Let H be a generalized Ffattini subgroup of G and let K be normal in G. If K∕H
is nilpotent, then K is nilpotent. Similarly, if the hypercommutator of K is contained
in H, then K is nilpotent. We consider the Fitting subgroup F(G) of a nonnilpotent
group G, and prove F(G) is a generalized Frattini subgroup of G if and only if every
solvable normal subgroup of G is nilpotent.
Now let H be a maximal generalized Frattini subgroup of a finite nonnilpotent
group G. Following Bechtell we introduce the concept of an H-series for G and prove
that if G possesses an H-series, then H = L(G).
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