Permutation representation contains trivial representation Let $G$ be a finite group and $H \vartriangleleft G$ a normal subgroup.
Let $(V,\rho)$ be the permutation representation (over $\mathbb{C})$ of $G$ acting on the set $G/H$ (we think of the quotient group as a set) in the natural way, i.e. for $s,t \in G$: $s \cdot (t\ (\textrm{mod}\ H)) = st\ (\textrm{mod}\ H)$.
Show that $(V,\rho)$ contains the trivial representation of $G$ with multiplicity 1.
My idea was to show that $G$ acts doubly transitively on $G/H$ and then apply this result.
But $G$ in particular need not act doubly transitively on itself, so we cannot apply this result.

Another way to approach this is with Frobenius Reciprocity and induction of characters. The permutation character equals $(1_H)^G$ and $[(1_H)^G, 1_G]=[1_H,1_H]=1$.
