Fixed point ratios in actions of finite classical groups, III

This is the third in a series of four papers on fixed point ratios for 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 for all elements x G of prime order, or (G,Ω) is one of a small number of known exceptions. In this paper we consider the case where Gω is contained in one of the Aschbacher families or .