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ALGORITHMS TO IDENTIFY ABUNDANT p-SINGULAR ELEMENTS IN FINITE CLASSICAL GROUPS

Published online by Cambridge University Press:  06 August 2012

ALICE C. NIEMEYER
Affiliation:
The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia (email: [email protected])
TOMASZ POPIEL
Affiliation:
The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia (email: [email protected])
CHERYL E. PRAEGER*
Affiliation:
The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia King Abdulaziz University, Jeddah, Saudi Arabia (email: [email protected])
*
For correspondence; e-mail: [email protected]
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Abstract

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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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

The first and third authors acknowledge the support of the Australian Research Council Discovery Project DP110101153. The second author acknowledges support within the Australian Research Council Federation Fellowship FF0776186 of the third author.

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