Minimal and random generation of permutation and matrix groups
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We prove explicit bounds on the numbers of elements needed to generate various types of finite permutation groups and finite completely reducible matrix groups, and present examples to show that they are sharp in all cases. The bounds are linear in the degree of the permutation or matrix group in general, and logarithmic when the group is primitive. They can be combined with results of Lubotzky to produce explicit bounds on the number of random elements required to generate these groups with a specified probability. These results have important applications to computational group theory. Our proofs are inductive and largely theoretical, but we use computer calculations to establish the bounds in a number of specific small cases.
Holt , D & Roney-Dougal , C M 2013 , ' Minimal and random generation of permutation and matrix groups ' , Journal of Algebra , vol. 387 , pp. 195-214 . https://doi.org/10.1016/j.jalgebra.2013.03.035
Journal of Algebra
© 2013 Elsevier Inc. This is the author’s version of a work that was accepted for publication in the Journal of Algebra. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Algebra, Vol 387, 2013. DOI: http://dx.doi.org/10.1016/j.jalgebra.2013.03.035
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