Vol. 23, No. 3, 1967

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Generalized Frattini subgroups of finite groups

James Clark Beidleman and Tae Kun Seo

Vol. 23 (1967), No. 3, 441–450

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).

Mathematical Subject Classification
Primary: 20.54
Received: 21 February 1967
Published: 1 December 1967
James Clark Beidleman
Tae Kun Seo