Algorithms to identify abundant p-singular elements in finite classical groups

Let G be a finite d-dimensional classical group and p a prime divisor of |G| distinct from the characteristic of the natural representation. We consider a subfamily of p-singular elements in G (elements with order divisible by p) that leave invariant a subspace X of the natural G-module of dimension greater than d/2 and either act irreducibly on X or preserve a particular decomposition of X into two equal-dimensional irreducible subspaces. We proved in a recent paper that the proportion in G of these so-called p-abundant elements is at least an absolute constant multiple of the best currently known lower bound for the proportion of all p-singular elements. From a computational point of view, the p-abundant elements generalise another class of p-singular elements which underpin recognition algorithms for finite classical groups, and it is our hope that p-abundant elements might lead to improved versions of these algorithms. As a step towards this, here we present efficient algorithms to test whether a given element is p-abundant, both for a known prime p and for the case where p is not known a priori.