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Please note the special place and time. This is a joint seminar with
Seminar on Reckoning.<br>
<br>
<div class="moz-cite-prefix">Dne 12.10.2025 v 10:00
<a class="moz-txt-link-abbreviated" href="mailto:kubis@math.cas.cz">kubis@math.cas.cz</a> napsal(a):<br>
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cite="mid:c83f97cd96c0e4ec4d899984f9d1c865@math.cas.cz">
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<title>Set Theory and Analysis</title>
<p> Wednesday, 15 October 2025 - 11:00 to 14:00 <br>
Place: blue lecture hall, ground floor, rear building </p>
<p> Speaker: Rosemary A. Bailey and Peter J. Cameron, University
of St Andrews<br>
Title: Permutation groups, lattices and orthogonal block
structures </p>
<p class="ql-ed"> Abstract <br>
</p>
<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>
<p> For more information see the seminar web page at <br>
<a class="moz-txt-link-freetext" href="https://www.math.cas.cz/index.php/events/seminar/6">https://www.math.cas.cz/index.php/events/seminar/6</a> </p>
<p> Set Theory and Analysis mailing list <br>
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<a class="moz-txt-link-freetext" href="https://list.math.cas.cz/listinfo/settfa@math.cas.cz">https://list.math.cas.cz/listinfo/settfa@math.cas.cz</a> </p>
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