Fixed point ratios in actions of finite classical groups, IV - Burness, T (Elsevier, 2007)
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)≲|xG|−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 Gω 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 Sp4(q)′ (q even) or PΩ+8(q). This completes the proof of the main theorem.