Probabilistic Generation of Wreath Products of Non-abelian Finite Simple Groups

Abstract

We show that the probability of generating an iterated standard wreath product of non-abelian finite simple groups is positive and tends to 1 as the order of the first simple group tends to infinity. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated. Information depending on the Classification of Finite Simple Groups is used throughout.

Key Words:

• Groups
• Generation
• Wreath product
• Probability

Mathematics Subject Classification:

• Primary 20F05
• Secondary 60B15, 20E22, 20E18

Acknowledgments

The author would like to thank Dr. John Bray, Dr. Christopher Parker and Professor Robert Wilson for their help and patience in many discussions relating to the subtleties of the non-abelian finite simple groups, and also for Dr. Parker's observation which greatly simplified the argument in Case (iii). The author would also like to thank Professor Dan Segal for drawing his attention to Professor Mann's article (Mann, 2004) and the referees of earlier drafts of this paper for their helpful comments.

Notes

^#Communicated by N. Gupta.