By definition, a finite group G is called an M-group if each of its irreducible complex characters is induced from a linear character of a subgroup of G. In this paper several theorems are proved. The most important are Theorem 1: Let G be an M-group and let S be a subnormal subgroup of odd index in G. Then every primitive character of S is linear. Theorem 2: Let G be an M-group, S a subnormal subgroup of G with \(| G:S|\) odd. Let \(\alpha \in Irr(S)\) be p-special for some prime p. Then \(\alpha\) is monomial. Theorem 1 follows from theorem 2 with the help of some lemmas and theorems also proved in this paper.