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A character of a group is said to be super-monomial if every primitivecharacter inducing it is linear. It is conjectured by Isaacs that everyirreducible character of an odd $M$-group is super-monomial. We show that allnon linear irreducible characters of lowest degree of an odd $M$-group issuper-monomial and provide cases in which one can guarantee that certainirreducible characters of normal subgroups are super-monomial. Finally, westudy groups having two irreducible monomial character degrees.

On super-monomial characters and groups having two irreducible monomial character degrees