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Generating simple groups and their subgroups

Banff International Research Station for Mathematical Innovation and Discovery

It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years, such as random generation, (2,3)-generation and so on. In this talk I will report on recent joint work with Martin Liebeck and Aner Shalev on similar problems for subgroups of (almost) simple groups, focussing on maximal and second maximal subgroups. We prove that every maximal subgroup of a simple group can be generated by four elements (this is best possible) and we show that the problem of determining a bound on the number of generators for second maximal subgroups depends on a formidable open problem in Number Theory. Time permitting, we will also present some related results on random generation and subgroup growth.

Mathematics; Group theory and generalizations; Topological groups, lie groups; Algebraic structures

video/mp4

Author affiliation: University of Bristol

2017-05-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61679

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/