## GR group theory – non-monomial groups with a special property

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.

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