A classical theorem of Jordan states that every finite transitive permutation group contains a derangement. This existence result has interesting and unexpected applications in many areas of mathematics, including graph theory, number theory and topology. Various generalisations have been studied in more recent years, with a particular focus on the existence of derangements with special properties. Written for academic researchers and postgraduate students working in related areas of algebra, this introduction to the finite classical groups features a comprehensive account of the conjugacy and geometry of elements of prime order. The development is tailored towards the study of derangements in finite primitive classical groups; the basic problem is to determine when such a group G contains a derangement of prime order r, for each prime divisor r of the degree of G. This involves a detailed analysis of the conjugacy classes and subgroup structure of the finite classical groups.

Let G be a permutation group on a finite set Ω. A base for G is a subset B ⊆ Ω with pointwise stabilizer in G that is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) 6 if G is an almost simple group of exceptional Lie type and Ω is a primitive faithful G-set. An important consequence of this result, when combined with other recent work, is that b(G) 7 for any almost simple group G in a non-standard action, proving a conjecture of Cameron. The proof is probabilistic and uses bounds on fixed point ratios.

Let G be a permutation group acting on a set Ω. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. We write b(G) for the minimal size of a base for G. We determine the precise value of b(G) for every primitive almost simple sporadic group G, with the exception of two cases involving the Baby Monster group. As a corollary, we deduce that b(G) 7, with equality if and only if G is the Mathieu group M 24 in its natural action on 24 points. This settles a conjecture of Cameron.

Let G be a finite almost simple classical group and let Ω be a faithful primitive non-standard G-set. A base for G is a subset B ⊆ Ω whose pointwise stabilizer is trivial; we write b(G) for the minimal size of a base for G. A well-known conjecture of Cameron and Kantor asserts that there exists an absolute constant c such thatc for all such groups G, and the existence of such an undetermined constant has been established by Liebeck and Shalev. In this paper we prove that either b(G) 4, or G = U 6 (2).2, Gω = U 4 (3).2 2 and b(G) = 5.The proof is probabilistic, using bounds on fixed point ratios.

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 ω is contained in a maximal non-subspace subgroup which lies in one of the Aschbacher families C i , where 4 i 8.