Primitive characters of subgroups of M-groups
If \(G\) is a monomial group, it has been conjectured that Hall subgroups of \(G\) are monomial. Recently, some idea is taking form: perhaps, even the Hall subgroup normalizers are monomial. Toward this stronger result we prove that the irreducible primitive characters of Hall subgroup normalizers of M-groups are linear. One of the reasons why M-groups are so difficult to understand is because, as shown by E. C. Dade, every solvable group is a subgroup of some M-group. The techniques used in this paper allow us to give a restriction on Dade's theorem: If \(G\) is a monomial group and \(H\) is a subgroup of \(G\) with \(\pi\)-index, then every primitive character of \(H\) has \(\pi\)-degree. For instance, as a consequence of this, \(\text{SL}(2,3)\) cannot have odd index in an M- group.
scientific article; zbMATH DE number 778955
scientific article