Growth and generation in SL₂(ℤ∕pℤ)

We show that every subset of SL 2 (Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL 2 (Z/pZ), every element of SL 2 (Z/pZ) can be expressed as a product of at most O((log p) c ) elements of A A -1 , where c and the implied constant are absolute.