Abstract

Special issue in honor of Laci Babai's 60th birthday We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic.

Highlights

  • A crucial task in designing algorithms to compute with matrix groups defined over finite fields is to recognise whether a given matrix group H is isomorphic to a finite classical group and, if so, to find an isomorphism with the natural representation of H

  • The isomorphism is defined by identifying appropriate elements of H with the standard generators of the natural representation. Algorithms which accomplish this task are generally referred to as constructive recognition algorithms by Kantor and Seress (2001)

  • We concentrate on the important special case when H is already given in its natural representation, by a set of d × d matrices over a finite field Fq

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Summary

Introduction

A crucial task in designing algorithms to compute with matrix groups defined over finite fields is to recognise whether a given matrix group H is isomorphic to a finite classical group and, if so, to find an isomorphism with the natural representation of H. We concentrate on the important special case when H is already given in its natural representation, by a set of d × d matrices over a finite field Fq (but we have not yet found appropriate elements of H that can serve as standard generators). Qk power to elements leaving invariant an αk-dimensional subspace without a 1-eigenvalue and having a (d−αk)-dimensional 1-eigenspace and in addition elements in Qpkpd power to elements acting irreducibly on the αk-dimensional subspace Membership in these sets can readily be determined algorithmically: for example, in the case of Qk, by examining the degrees of the irreducible factors of the characteristic polynomial.

Eigenvalues of elements in maximal tori
The elements we seek
The ingredients of the proofs
Relevant conjugacy classes
Proportions of elements in tori which lie in Q
Proportions of elements in symmetric groups

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