The considered groups are finite and the characters are over $ mathbb $, A group $ G $ is monomial, if any irreducible character $ chi $ is induced by a linear character of a subgroup $ H subsqqq $, Of course, there are many groups that are not monomial (for example, any group that is not solvable).
A generalization of monomial groups is that with the property that it stands for any irreducible character $ chi $there is an integer $ e> 0 $ so that $ e chi $ is induced by a linear character of a subgroup $ H subsqqq $,
My question is the following: Let $ G $ be an arbitrary group. Two different, irreducible characters $ chi $ and $ psi $ from $ G $It is always possible to find a subgroup $ H subsqqq $ and a linear character $ varphi $ so that $ langle varphi ^ G, chi rangle neq 0 $ and $ langle varphi ^ G, psi rangle = 0 $, from where $ varphi ^ G $ is the induced sign up $ G $?
Of course, the (generalized) monomial groups fulfill this property, since the irreducible signs form an orthonomic basis in the space of class functions.