BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Peter Cameron (University of St Andrews) DTSTART:20200522T080000Z DTEND:20200522T090000Z DTSTAMP:20260404T110831Z UID:GroupsAndCombinatoricsSeminar/1 DESCRIPTION:Title: The geometry of diagonal groups\nby Peter Cameron (University of St Andrews) as part of Groups and Combinatorics Se minar\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/GroupsAndCombinatoricsSe minar/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Eric Swartz (College of William and Mary) DTSTART:20200604T120000Z DTEND:20200604T130000Z DTSTAMP:20260404T110831Z UID:GroupsAndCombinatoricsSeminar/2 DESCRIPTION:Title: Fuchs' problem for 2-groups\nby Eric Swar tz (College of William and Mary) as part of Groups and Combinatorics Semin ar\n\n\nAbstract\nWe say that a group G is realizable if there exists a ri ng R such that\nthe group of units of R is isomorphic to G. Sixty years a go\, László\nFuchs posed the problem of determining which groups are rea lizable as\nthe group of units of a commutative ring\, and the question of \ndetermining whether a group or family of groups is realizable in any\nri ng has come to be called Fuchs' problem. In recent years\, Fuchs'\nproble m has been studied for various families of groups\, such as\ndihedral grou ps and simple groups\, although the problem of determining\nprecisely whic h groups are realizable is still very open in general\n(and is even still open in the case when the ring is commutative\, as\nin Fuchs' original que stion). In this talk\, we will consider the\nquestion of which 2-groups a re realizable as unit groups of finite\nrings\, a necessary step toward de termining which nilpotent groups are\nrealizable. This is joint work with Nicholas Werner.\n\nZoom link available 8 hrs before talk. See Eric's rec ent Journal of Algebra paper (with the same title). This seminar is at an unusual time because Eric will be speaking from Virginia\, USA.\n LOCATION:https://stable.researchseminars.org/talk/GroupsAndCombinatoricsSe minar/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Dr G. Luke Morgan (FAMNIT\, University of Primorska) DTSTART:20200626T080000Z DTEND:20200626T090000Z DTSTAMP:20260404T110831Z UID:GroupsAndCombinatoricsSeminar/3 DESCRIPTION:Title: Some small progress on the PSV Conjecture \nby Dr G. Luke Morgan (FAMNIT\, University of Primorska) as part of Group s and Combinatorics Seminar\n\n\nAbstract\nThe subject of the talk is the question of bounding the number of automorphisms of arc-transitive graphs in terms of the valency of the graph. More specifically\, we consider the question for groups acting arc-transitively on graphs such that the local action (that induced on the neighbours of a vertex by the stabiliser of th at vertex) is semiprimitive. This question was originated by Weiss for the case of primitive local action and generalised by Praeger for the case of quasiprimitive local action. I will report on some recent small progress on the first type - that of semiprimitive local action. The result is akin to Tutte’s famous result on cubic s-arc transitive graphs where the num ber of automorphisms is bounded by 3*2^(s-1). Tutte's proof was elegant\, elementary and self-contained. The recent progress relies on some group th eoretical tools that were developed for use in the Classification of the F inite Simple Groups - and some tricks to allow us to patch things together . I'll try to present these results in a friendly fashion\, as well as kee ping in mind the ``big picture'' concerning where progress now stands on t hese conjectures.\n LOCATION:https://stable.researchseminars.org/talk/GroupsAndCombinatoricsSe minar/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Gareth Tracey (Renyi Institute\, Budapest) DTSTART:20200703T080000Z DTEND:20200703T090000Z DTSTAMP:20260404T110831Z UID:GroupsAndCombinatoricsSeminar/4 DESCRIPTION:Title: On the Chebotarev invariant of a finite group \nby Gareth Tracey (Renyi Institute\, Budapest) as part of Groups and Combinatorics Seminar\n\n\nAbstract\nGiven a finite group X\, a classical approach to proving that X is the Galois group of a Galois extension K/Q c an be described roughly as follows: (1) prove that Gal(K/Q) is contained i n X by using known properties of the extension (for example\, the Galois g roup of an irreducible polynomial f (x) ∈ Z[x] of degree n embeds into t he symmetric group Sym(n))\; (2) try to prove that X = Gal(K/Q) by computi ng the Frobenius automorphisms modulo successive primes\, which gives conj ugacy classes in Gal(K/Q)\, and hence in X. If these conjugacy classes can only occur in the case Gal(K/Q) = X\, then we are done. The Chebotarev in variant of X can roughly be described as the efficiency of this “algorit hm”. In this talk we will define the Chebotarev invariant precisely\, an d describe some new results concerning its asymptotic behaviour.\n LOCATION:https://stable.researchseminars.org/talk/GroupsAndCombinatoricsSe minar/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Saul Freedman (University of St Andrews) DTSTART:20200731T080000Z DTEND:20200731T090000Z DTSTAMP:20260404T110831Z UID:GroupsAndCombinatoricsSeminar/5 DESCRIPTION:Title: The non-commuting\, non-generating graph of a group\nby Saul Freedman (University of St Andrews) as part of Groups and Combinatorics Seminar\n\n\nAbstract\nGiven a group G\, we can construc t associated graphs that encode certain relations between the elements (or subgroups) of G. A well-known example is the generating graph of G\, whos e vertices are the nontrivial elements of G\, with two vertices joined if the elements form a generating set for G. In June this year\, Burness\, Gu ralnick and Harper showed that if the generating graph of a finite group h as no isolated vertices\, then it as "dense" as possible\, in the sense th at it is connected with diameter at most 2. This generalises a famous resu lt of Breuer\, Guralnick and Kantor from 2008: the generating graph of a n on-abelian finite simple group is connected with diameter 2.\n\nConsider n ow the non-commuting\, non-generating graph of G\, obtained by taking the complement of the generating graph\, removing edges between elements that commute\, and finally removing vertices corresponding to elements of Z(G). In this talk\, we explore the connectedness and diameter of this graph fo r finite (and certain infinite) groups G\, for example by studying the max imal subgroup structure of G. In particular\, we prove a result that is pe rhaps surprising: in many cases\, this naturally-defined subgraph of the c omplement of the dense generating graph is itself similarly dense.\n\nWe a lso present in this talk a new upper bound on the diameter of a related gr aph: the intersection graph of a finite non-abelian simple group. The vert ices of this graph are the nontrivial proper subgroups of the group\, with two subgroups joined if they intersect nontrivially.\n\nPassword hint: 04 7877+ the order of Alt(5)\n LOCATION:https://stable.researchseminars.org/talk/GroupsAndCombinatoricsSe minar/5/ END:VEVENT END:VCALENDAR