  We prove that Boolean functions on $S_{n}$ whose Fourier transform is highly
concentrated on the first two irreducible representations of $S_n$, are close
to being unions of cosets of point-stabilizers. We use this to give a natural
proof of a stability result on intersecting families of permutations,
originally conjectured by Cameron and Ku, and first proved by the first author.
We also use it to prove a `quasi-stability' result for an edge-isoperimetric
inequality in the transposition graph on $S_n$, namely that subsets of $S_n$
with small edge-boundary in the transposition graph are close to being unions
of cosets of point-stabilizers.
