Representations and characters of the induced representation The following theorem is in my lecture notes, without proof:

Let H be a subgroup of G and let $r : H \to GL(V)$ be the trivial
  representation of $H$. Then $Ind^G_H(V) \cong C[G/H]$, where $C[G/H]$
  is the permutation representation of $G$ corresponding to the natural
  action of $G$ on the set of left cosets of $H$ in $G$.

I am trying to prove this. I believe the best approach is to compare characters, but I don't quite see how we get equal characters here. Any hints would be appreciated.
Update: I think Frobenius reciprocity will help.
 A: You're probably better off working directly from the definition of the induced representation.
In general, if $r : H \to Gl(V)$ is a representation of a subgroup $H \subset G$, then the induced representation ${\rm Ind}_H^G  r$ is defined as follows. First, we fix representatives $\{ g_1, \dots, g_n \}$ of the cosets of $H$, and we take the direct sum of $n$ isomorphic copies of $V$, labelling them with the representatives of the cosets:
$$ W := g_1V \oplus g_2V \oplus \dots \oplus g_n V$$
We then define the induced representation ${\rm Ind}_H^G  r$ as a representation on this direct sum $W$. For any $g \in G$ and for any coset representative $g_i$, one can find a unique coset representative $g_j$ and a unique element $h \in H$ such that $gg_i = g_j h$. We then define the action of $g$ on an element $g_i v \in g_i V$ as follows:
$$ {\rm Ind}_H^G  r(g)(g_i v) = g_j(r(h)v) \in g_j V.$$
This definition is extended by linearity to the whole of $W$.
In your example, $V = \mathbb C$, so $V$ is spanned by a single basis vector, which we denote by $e$. For each coset representative $g_i$, we denote the corresponding basis vector of $g_i V$ by $e_{g_i}$. We know that $r$ is the trivial representation, so $r(h) e_{g_i} = e_{g_i}$ for any coset representative $g_i$ and for any $h \in H$. Therefore, for any $g \in G$,
$$ {\rm Ind}_H^G r(g)(e_{g_i}) = e_{g_j},$$
where $g_j$ is the coset representative such that $gg_i \in g_j H,$ i.e. such that $g(g_i H) = g_j H$. And this is precisely the permutation representation corresponding to the natural action of $G$ on the cosets $\{ g_1 H, \dots, g_n H \}$.
