Cosets of Sylow p-subgroups and a question of Richard Taylor

We prove that for any odd prime p, there exist infinitely many finite simple groups S containing a Sylow p-subgroup P of S such that there exists a coset of P in S with the property that all elements in the coset have order a multiple of p. This allows us to answer a question of Richard Taylor related to whether certain Galois representations are automorphic.