If $G$ is a finite solvable group, it is known (for example, Murty & Raghuram 2000, lemma 2.4) that $Ind\_H^G 1\_H-1\_G$ can be expressed directly (with all coefficients $=1$) as a sum of monomial characters.


What about $G$ non-solvable?


I guess it is one of two options: 1) it is known to be false, 2) the same is expected to be true but not known.


I'd appreciate any information or reference in either direction, or any partial result on $Ind\_H^G 1\_H-1\_G$ that improves on Brauer's theorem.



$Ind_H^G 1_H-1_G$ as direct sum of monomials
It does not hold in general: For instance take $G=S\_5$, the symmetric group on $5$ letters, and $H=S\_4$. Then $\text{Ind}\_H^G1\_H-1\_G$ is an irreducible character of degree $4$ (by $2$-transitivity of $S\_5$). Thus if it were a sum of monomial characters, it would have to be monomial, implying that $S\_5$ has a subgroup of index $4$, which of course isn't the case.


