Let $H < G$ be finite groups with $|G:H|=n$, and let $M$ be an irreducible $FH$-module for some field $F$. Is it always true that all irreducible constituents of the induced $FG$-module $M^G$ have dimension at least $\dim M$? If not, then is it true that the composition length of $M^G$ is at most $n$?


For complex representations, these statements follow from Frobenius Reciprocity. But do they remain true for modular representations? I am particularly interested in the case when $F$ is a finite prime field.



Constituents of induced representation
I am assuming you mean "compositon factor" when you speak of irreducible constituent. In the algebraically closed case you seem to be asking the following question in the first part, phrased in terms of Brauer characters: Let $\phi$ be a Brauer character of $H,$ and $\psi$ be a Brauer character of $G.$ Let $\alpha$ be the Brauer character of the projective indecomposable of $H$ corresponding to $\phi$ and $\beta$ be the Brauer character of the projective indecomposable of $G$ corresponding to $\psi.$ Can we have $\langle \beta|\_{H}, \phi \rangle \neq 0$ when $\psi(1) < \phi(1)?$ Put this way, I don't immediately see why not, so I think the answer to the first question may be no, but that is not a proof, and I may have missed something.


