Constituents of induced representation 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.
 A: 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.
A: In the case that $n$ is prime and $H$ is normal in $G$. Your second questions has an affirmative answer. The composition length is $1$, $n$, or $1+d$ where $d\mid (n-1)$, in each case at most $n$. See the main theorem from:
S.P. Glasby and L.G. Kovács,
Irreducible modules and normal subgroups of prime index.
Comm. Algebra 24 (1996), no. 4, 1529–1546.
A: Write $M^G$ as a direct sum of translated copies of $M$, one for each coset in $G/H$, that are permuted by the action of $G$. This defines a projection $\varphi : M^G \to M$ that looks only at the copy corresponding to the trivial coset. This projection is $H$-equivariant, so the image of an irreducible subrepresentation of $M^G$ is either $0$ or all of $M$. 
Let $v$ be a nonzero element of the irreducible subrepresentation. Then it must have a nonzero projection onto the copy of $M$ corresponding to some coset, $gH$. Then $g^{-1} v$ has a nontrivial projection onto the copy of $M$ corresponding to  $H$, so the image of $\varphi$ is not $0$, so it is all of $M$, so the dimension is at least $\operatorname{dim} M$.
This method only gets irreducible subrepresentations, not composition factors.
