Frobenius reciprocity and induced representations In representation theory, we consider the restriction functor for any group $G$ and subgroup $H$. This is:
$Res_H^G : Rep(G) \rightarrow Rep(H)$
This gives a representation of $H$
The Induced case lets us look in the other direction, going from subgroup representations to the groups representation
$Ind_H^G : Rep(H) \rightarrow Rep(G)$
How does this relate to Frobenius Reciprocity?
For example, I have the following question that I would like to be able to solve

Let $G$ be a finite group and $H$ be a subgroup. Let $1_H$ be the trivial representation of $H$. Show that the trivial representation $1_G$ of $G$ occurs exactly once in the induced representation $Ind_H^G1_H$

My instinct would be to assume $1_G$ appears more than once in the induced rep, and arrive at a contradiction. I have not yet been sucessful in this and would very much appreciate your help
 A: Frobenius Reciprocity says that induction and restriction are adjoint functors.  If you aren't comfortable with that language, what this means is that if $V$ is a representation of $H$ and $W$ is a representation of $G$ then there is an isomorphism:
$$Hom_G(Ind_H^G (V), W) \cong Hom_H(V, Res^G_H (W))  $$
Really adjunction gives something stronger, a sort of consistent way to choose such an isomorphism for all such pairs $V$ and $W$, but for a lot of purposes this statement is what we really care about.
Applying this to your question, where $V = 1_H$ and $W = 1_G$, we get:
$$Hom_G(Ind_H^G (1_H), 1_G) \cong Hom_H(1_H, Res^G_H (1_G))  $$
The point is that while the left hand side involves a term $Ind_H^G (1_H)$ we don't quite understand, the left hand side is easy since clearly $Res^G_H (1_G) = 1_H$. So this just becomes
$$Hom_G(Ind_H^G (1_H), 1_G) \cong Hom_H(1_H, 1_H) \cong \mathbb{C}$$
So this tells us that $Hom_G(Ind_H^G (1_H), 1_G)$ is one dimensional, which exactly means that there is a single copy of the trivial representation of  $G$ in $Ind_H^G (1_H)$, since if there were two copies we could project onto each one and scale them independently giving us a(n at least) two dimensional Hom-space.
