Die Praefrattinigruppe im Intervall eines Untergruppenverbandes. (The prefrattini group in the interval of a subgroup lattice) (Q1824695)

Let H be a proper subgroup of finite solvable group G, let [H,G] be the set of all subgroups of G containing H, and let max [H,G] be the set of maximal subgroups of G which contain H. Let \(1=N_ 0<...<N_ n\) be a chief series of G and for each i set \(M_ i=\{M\in \max [H,G]:\) \(MN_ i=G\) and \(M\cap N_ i=N_{i-1}\}\) or \(\emptyset\) if no such M exists. Denote by D[H,G] the set of intersections of each set of subgroups, one from every nonempty \(M_ i\). By Corollary 2.4 D[H,G] is independent of the particular chief series used. The elements of D[H,G] are the H- prefrattini groups of G, and the author obtains analogs of some of the results of W. Gaschütz on prefrattini groups. In particular (Theorem 3.4) for each H the H-prefrattini groups form a conjugate class in G, and (Theorem 3.3) H-prefrattini groups are reasonably well-behaved under homomorphisms and normal intersections.