Primitive characters of subgroups of $M$-groups

Abstract

Let G be an M-group, let S be a subnormal subgroup of G, and let H be a Hall subgroup of S. If the character a e Irr(H) is primitive, then 'y(l) is a power of 2. Furthermore, if IG: SI is odd, then -y(I) = 1. Combining Theorems A of (5) and (13), Isaacs has proved that if G is an M- group and S is a subnormal subgroup of G, then the primitive characters of S all have degrees that are powers of 2, and if S has odd index in G, then the primitive characters of G are all linear. Using the techniques found in (12) and (13), we have