On proportions of pre-involutions in finite classical groups

Key to a computational study of the finite classical groups in odd characteristic are efficient methods for constructing involutions and their centralisers. Constructing an involution in a given conjugacy class is usually achieved by finding an element of even order that powers up to an involution in the class. Lower bounds on the proportions of such elements are therefore required to control the failure probability of these algorithms. Previous results of Christopher Parker and Robert Wilson give an O(n−3) lower bound that holds for all involution classes in n-dimensional simple classical groups in odd characteristic. We improve this lower bound to , and in certain cases to O(n−1), and also treat a larger family of (not necessarily simple) classical groups.