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SUMMARY; CHARSET=UTF-8 :NTAG Seminar: The number and nature of subgroups of the symmetric group
UID:exeter_event_14548
URL:http://www.exeter.ac.uk/events/details/?event=14548
DTSTART;VALUE=DATE:20250122T143500
DTEND;VALUE=DATE:20250122T152500
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ORGANIZER: MAILTO:c.d.lazda@exeter.ac.uk
ATTACH: http://www.exeter.ac.uk/events/details/?event=14548
DTSTAMP:20250109T093200
LOCATION:Harrison Building 106 
DESCRIPTION; CHARSET=UTF-8 :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.

New results in this talk are joint w/ Gareth Tracey.http://www.exeter.ac.uk/events/details/?event=14548
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