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The symmetries of any object are described by a group, so it is natural to ask: What does a random group look like? This talk will start with a brief survey of how we might go about counting various algebraic structures. We'll then go on to see what a random group might be, in various different contexts.
A symmetric group on some set \( \Omega \) is the group of all permutations of \( \Omega \), under composition of functions. Every group arises as a subgroup of some symmetric group, so fully understanding the symmetric group means understanding all groups. An elementary argument shows that there are at least \( 2^{n^2/16} \) subgroups of a symmetric group on \( n \) points, and it was conjectured by Pyber in 1993 that up to lower order error terms this is also an upper bound. The same year, Kantor conjectured that a random subgroup of the symmetric group is nilpotent. This talk will present a proof of one of these conjectures, and a disproof of the other.
The new results in this talk are joint work with Gareth Tracey (Warwick).
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