Characters of Maximal Subgroups ofM-Groups

Abstract

In this paper, we will prove some results about the structure of M-groups. We want to study the degrees of primitive characters of subgroups of M-groups. Since the subgroups of M-groups need not themselves be M-groups, we know that the primitive characters of the subgroup need not be linear, and so, we want to study the relationship between the degrees of the primitive characters and the index of the subgroups. This line of study w x is motivated by Theorem B of 20 , where G. Navarro proved that if G is an M-group and H is a subgroup with p-index for some set of primes p , then every primitive character of H has p-degree. In this paper, we will be able to get a stronger relationship between the degrees of primitive characters and the indices of subgroups of M-groups. The result of Navarro led us to conjecture that if G is an M-group and H is any subgroup, then the degrees of all the primitive irreducible characters in H must divide the index of H in G. However, almost immediately after making this conjecture, we noticed that the group that w x Dade constructed in 1 is a counterexample to this conjecture. This example relies heavily on the properties of the prime 2, and so we decided to check the validity of the conjecture when we make an additional oddness assumption. When the group has a maximal subgroup of odd index, we have discovered the following result.