Derangements in finite classical groups and characteristic polynomials of random matrices

Abstract: We first obtain explicit upper bounds for the proportion of elements in a finite classical group G with a given characteristic polynomial. We use this to complete the proof that the proportion of elements of a finite classical group G which lie in a proper irreducible subgroup tends to 0 as the dimension of the natural module goes to infinity. This result is analogous to the result of Luczak and Pyber [15] that the proportion of elements of the symmetric group S_n which are contained in a proper transitive subgroup other than the alternating group goes to 0 as n goes to infinity. We also show that the probability that 3 random elements of SL(n,q) invariably generate goes to 0 as n goes to infinity.