I did my [M.Sc thesis][1] on random walks on finite groups and you might be interested in the proof of Theorem 1.3.2 in particular.

When I was looking for the proof of this theorem, almost all references refer
to the proof in older hard-to-source references — if at all. A proof outline is given
by Fountoulakis in his lecture notes but here a full proof is given.


  [1]: http://jpmccarthymaths.files.wordpress.com/2010/09/current.pdf