ALGORITHMS TO IDENTIFY ABUNDANTp-SINGULAR ELEMENTS IN FINITE CLASSICAL GROUPS

Abstract

AbstractLetGbe a finited-dimensional classical group andpa prime divisor of ∣G∣ distinct from the characteristic of the natural representation. We consider a subfamily ofp-singular elements inG(elements with order divisible byp) that leave invariant a subspaceXof the naturalG-module of dimension greater thand/2 and either act irreducibly onXor preserve a particular decomposition ofXinto two equal-dimensional irreducible subspaces. We proved in a recent paper that the proportion inGof these so-calledp-abundantelements is at least an absolute constant multiple of the best currently known lower bound for the proportion of allp-singular elements. From a computational point of view, thep-abundant elements generalise another class ofp-singular elements which underpin recognition algorithms for finite classical groups, and it is our hope thatp-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 isp-abundant, both for a known primepand for the case wherepis not knowna priori.