Fixed point ratios, Sylow numbers and coverings of p-elements in finite groups

Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime p and a finite group G, we use fixed point ratios to study the number of Sylow p-subgroups of G and the minimal size of a covering by proper subgroups of the set of p-elements of G.