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Wednesday, 15 October 2025 - 11:00 to 14:00 <br />
Place: blue lecture hall, ground floor, rear building
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Speaker: Rosemary A. Bailey and Peter J. Cameron, University of St Andrews<br />
Title: Permutation groups, lattices and orthogonal block structures
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Abstract <br />
<p>The story began when our coauthor Marina was doing an undergraduate research internship under Peter's supervision. We were studying transitive but imprimitive permutation groups through their invariant equivalence relations, and were looking at the case where the equivalence relations commute; in our shared office, Rosemary overheard our conversation, and said, "Statisticians know about those things; we call them orthogonal block structures."</p><p>An orthogonal block structure (OBS) is a lattice of commuting uniform equivalence relations. These are combinatorial objects which may have trivialautomorphism group. Latin squares provide many examples of OBSs. We will discuss the history of how OBSs arose in experimental design.</p><p>A better behaved special case occurs when the lattice is distributive; these are called poset block structures. They always have a large automorphism group, a generalised wreath product of symmetric groups, described by a poset with a set attached at each of its points. Our main results are a proof that a group preserving a poset block structure is contained in a generalised wreath product of permutation groups defined from the action (an extension of the Krasner-Kaloujnine theorem), and that a generalised wreath product over a poset is the intersection of the iterated wreath products of the same groups over all linear extensions of the poset.</p><p>In terms of the group operation, so that the graph is invariant under all automorphisms of the group. The classical example is the commuting graph of a group, defined by Brauer and Fowler in a seminal paper in 1955.</p><p>There has been a lot of work on this topic recently. My interests are mainly in how the theories of groups and graphs can help one another. Some of the questions I will address are</p><p> - finding new results about groups;</p><p> - characterising important classes of groups using graphs;</p><p> - recognising graphs obtained from groups and, if possible, reconstructing the groups;</p><p> - finding some beautiful graphs.</p><p>Recently I have widened the investigation to simplicial complexes defined on groups; I will present a small amount of new material and some open problems on this also. </p><p><br></p>
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For more information see the seminar web page at <br />
https://www.math.cas.cz/index.php/events/seminar/6
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Set Theory and Analysis mailing list <br />
settfa@math.cas.cz <br />
https://list.math.cas.cz/listinfo/settfa@math.cas.cz
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