# The Orders of Simple Groups

The Classification of Finite Simple Groups proves in particular that the collection of orders of finite simple groups has asymptotic density 0. What can we prove if we’re not willing to work that hard?

Landau considered the following question – What is the maximal order of an element in the permutation group on k letters? We’ll prove some elementary bounds and deduce an asymptotic using the PNT.

Does there exist a pair of loaded six-sided dice such that the probability of rolling any dice sum {2,..12} is equally likely? We’ll show how this and other related questions about dice sums can be analyzed using cyclotomic polynomials.