HIMR-sponsored Algebra seminar - Colva Roney-Dougal
| Dates: | 13 May 2025 |
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| Times: | 14:00 - 15:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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| Speaker: | Colva Roney-Dougal |
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Title: Asymptotic enumeration of permutation groups
Abstract: One of the most elementary, but difficult, questions we can ask about a finite group G is how many subgroups it has. An elementary argument shows that the symmetric group on n points has at least 2^{n^2/16} subgroups. Pyber showed in 1993 that it has at most 24^{n^2/6 + o(n^2)} subgroups, and conjectured that the elementary lower bound is, up to error terms, the correct upper bound. This talk will present some ideas from our recent proof of this conjecture.
One reason for enumerating these groups is to determine properties of randomly-chosen permutation groups. Erdos conjectured that if m \leq 2^a then the number of groups of order m is bounded above by the number of groups of order 2^a, and building on this Pyber conjectured in 1993 that as m tends to infinity the probability that a random group of order at most m is nilpotent tends to 1. In a similar vein, Kantor conjectured in 1993 that the probability that a random subgroup of the symmetric group on n points is nilpotent tends to 1. I will show that one of these conjectures is false.
Joint work with Gareth Tracey (Warwick).
Speaker
Colva Roney-Dougal
Organisation: University of St Andrews
Travel and Contact Information
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Frank Adams 1
Alan Turing Building
Manchester
