Fixed point ratios in actions of finite classical groups, II

This is the second in a series of four papers on fixed point ratios in non-subspace actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G-set then either fpr(x )<|x^G|^{−1/2} for all elements x ∈ G of prime order, or (G, Ω) is one of a small number of known exceptions. In this paper we record a number of preliminary results and prove the main theorem in the case where the stabiliser G_{a} is contained in a maximal non-subspace subgroup which lies in one of the Aschbacher families C_i , where i = 4,5,6,7 or 8.