Finite groups with nilpotent commutator subgroups, having a distributive lattice of normal subgroups
The authors are concerned with a finite group G whose lattice, n(G), of normal subgroups is distributive. First they show that if G' is nilpotent then n(G) is distributive if and only if every epimorph of G has a faithful irreducible character; in fact, if G is solvable and n(G) is distributive then every epimorph of G has a faithful irreducible character. An example is given of a group G with G' supersolvable for which every epimorph has a faithful irreducible character but n(G) is not distributive. Then the distributivity of n(G) when G' is nilpotent is reduced to the distributivity of n(H) for certain subgroups H of G. This means that the distributivity of n(G) for G supersolvable reduces to the strictly p-closed case; distributivity in this case is characterized in Section 2. Then in Section 3 a nonsupersolvable group G is constructed, using the wreath product, with G' nilpotent and n(G) distributive.
distributive lattice of normal subgroups
scientific article; zbMATH DE number 3944135
scientific article