[Settfa] [Set Theory]Wednesday 15 Oct, Rosemary A. Bailey and Peter J. Cameron

[Adam Bartoš]

Please note the special place and time. This is a joint seminar with 
Seminar on Reckoning.

Dne 12.10.2025 v 10:00 kubis at math.cas.cz napsal(a):
> Set Theory and Analysis
>
> Wednesday, 15 October 2025 - 11:00 to 14:00
> Place: blue lecture hall, ground floor, rear building
>
> Speaker: Rosemary A. Bailey and Peter J. Cameron, University of St Andrews
> Title: Permutation groups, lattices and orthogonal block structures
>
> Abstract
>
> 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."
>
> 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.
>
> 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.
>
> 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.
>
> 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
>
>  - finding new results about groups;
>
>  - characterising important classes of groups using graphs;
>
>  - recognising graphs obtained from groups and, if possible, 
> reconstructing the groups;
>
>  - finding some beautiful graphs.
>
> 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.
>
>
> For more information see the seminar web page at
> https://www.math.cas.cz/index.php/events/seminar/6
>
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