Erd\H{o}s' Multiplication Table Problem for Function Fields and Symmetric Groups

Erd\H{o}s first showed that the number of positive integers up to $x$ which can be written as a product of two number less than $\sqrt{x}$ has zero density. Ford then found the correct order of growth of the set of all these integers. We will use the tools developed by Ford to answer the analogous question in the function field setting. Finally, we will use a classical result relating factorization of polynomials to factorization of permutations to recover a result of Eberhard, Ford and Green of an analogous multiplication table problem for permutations.