Save Big on Coursera Plus. 7,000+ courses at $160 off. Limited Time Only! Grab it Explore advanced mathematical concepts in this 47-minute lecture focusing on two key results concerning symmetric groups. Delve into the characterization of subsets A of An whose square A^2 covers the entire alternating group An, discovering how conjugacy classes with density of at least exp(-n^{2/5} - ε) lead to An = A^2 - an improvement on Larsen and Shalev's previous findings. Examine the Erdős-Sós forbidden intersection problem for permutation families, learning how non-(t-1)-intersecting families with t ≤ √n/log(n) have a maximal size of (n-t)!. Understand how both results connect to independent sets in normal Cayley graphs over symmetric groups, utilizing hypercontractive inequalities for global functions developed by Keller, Lifshitz, Marcus, Keevash, and others.