On faithful permutation representations of small degree

We study the obstacles for a family of finite groups to having faithful permutation representations of arbitrarily small degrees compared to their respective orders. We show that the only obstacle is that each group has a cyclic subgroup of prime-power order and bounded index. Specifically, we show the following: Let μ(G) be the smallest degree of a faithful permutation representation of G, and let i(G) be the index of its largest cyclic subgroup of prime-power order. Then for a constant c > 0