  This is an essay about a certain family of elements in the general linear
group GL(d,q) called primitive prime divisor elements, or ppd-elements. A
classification of the subgroups of GL(d,q) which contain such elements is
discussed, and the proportions of ppd-elements in GL(d,q) and the various
classical groups are given. This study of ppd-elements was motivated by their
importance for the design and analysis of algorithms for computing with matrix
groups over finite fields. An algorithm for recognising classical matrix
groups, in which ppd-elements play a central role is described.
