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

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  06 August 2012

ALICE C. NIEMEYER ,
TOMASZ POPIEL  and
CHERYL E. PRAEGER
ALICE C. NIEMEYER
Affiliation:
The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia (email: alice.niemeyer@uwa.edu.au)
TOMASZ POPIEL
Affiliation:
The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia (email: tomasz.popiel@uwa.edu.au)
CHERYL E. PRAEGER*
Affiliation:
The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia King Abdulaziz University, Jeddah, Saudi Arabia (email: cheryl.praeger@uwa.edu.au)
*
✉For correspondence; e-mail: cheryl.praeger@uwa.edu.au
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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.

Keywords

finite groups of Lie typeproportions of elementslarge invariant subspaces

MSC classification

Secondary: 20G40: Linear algebraic groups over finite fields
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 1 , August 2012 , pp. 50 - 63
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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