✦ LIBER ✦

Cyclic Matrices in Classical Groups over Finite Fields

✍ Scribed by Peter M. Neumann; Cheryl E. Praeger

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 362 KB
Volume: 234
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.2000.8548

No coin nor oath required. For personal study only.

✦ Synopsis

to helmut wielandt for his 90th birthday with much respect and many congratulations

, where m X t is its minimal polynomial and c X t is its characteristic polynomial det tI -X . This condition is equivalent to requiring the vector space F d of 1 × d row vectors over F to be cyclic as an F X -module. In a previous paper we showed that most d × d matrices over a finite field F are cyclic. The present work is a continuation of that. Its aim is to obtain good lower bounds on the proportion of cyclic matrices in the general linear group GL d F and in various important subgroups of it. Although our motivation originated in our work on the design and analysis of algorithms for computing efficiently in matrix groups, the results have turned out to be of independent interest.

Define Cyc d q to be the set of cyclic matrices in M d q , and define Noncyc d q to be the set of non-cyclic matrices. The proportion Noncyc d q ÷ q d 2 may be naturally thought of as the probability that a randomly chosen d × d matrix is not cyclic. In we proved that Prob X ∈ M d q is non-cyclic = q -3 + O q -4

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