Fixed point ratios in actions of finite classical groups, IV

This is the final paper in a series of four 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 assume G_{a} is either an almost simple irreducible subgroup in Aschbacher’s S collection, or a subgroup in a small additional set N which arises when G has socle Sp(4,q)′ (q even) or PΩ(+,8,q). This completes the proof of the main theorem.