A subset of a group is said to be product free if it does not contain the product of two elements in it. We consider how large can a product free subset of the alternating group An be?   In the talk we will completely solve the problem by determining the largest product free subset of An. Our proof combines a representation theoretic argument due to Gowers, with a new analytic tool called hypercontractivity for global functions.   Based on a joint work with Peter Keevash and Dor Minzer