Minimal and random generation of permutation and matrix groups

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 ge...

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Bibliographic Details
Published inJournal of algebra Vol. 387; pp. 195 - 214
Main Authors Holt, Derek F., Roney-Dougal, Colva M.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.08.2013
Subjects
Finite groups
Generation of groups
Matrix groups
Permutation groups
probabilistic and asymptotic group theory
20P05
Matrix groups
Generation of groups
Permutation groups
Finite groups
probabilistic and asymptotic group theory
20B05
20H20
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Summary: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.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2013.03.035