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:20260422T225725Z UID:GroupsAndCombinatoricsSeminar/1 DESCRIPTION:Title: The geometry of diagonal groups\nby Peter Camero n (University of St Andrews) as part of Groups and Combinatorics Seminar\n \nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/GroupsAndCombinatoricsSeminar/1 / END:VEVENT BEGIN:VEVENT SUMMARY:Eric Swartz (College of William and Mary) DTSTART:20200604T120000Z DTEND:20200604T130000Z DTSTAMP:20260422T225725Z UID:GroupsAndCombinatoricsSeminar/2 DESCRIPTION:Title: Fuchs' problem for 2-groups\nby Eric Swartz (Col lege of William and Mary) as part of Groups and Combinatorics Seminar\n\n\ nAbstract\nWe say that a group G is realizable if there exists a ring R su ch that\nthe group of units of R is isomorphic to G. Sixty years ago\, L ászló\nFuchs posed the problem of determining which groups are realizabl e as\nthe group of units of a commutative ring\, and the question of\ndete rmining whether a group or family of groups is realizable in any\nring has come to be called Fuchs' problem. In recent years\, Fuchs'\nproblem has been studied for various families of groups\, such as\ndihedral groups and simple groups\, although the problem of determining\nprecisely which grou ps are realizable is still very open in general\n(and is even still open i n the case when the ring is commutative\, as\nin Fuchs' original question) . In this talk\, we will consider the\nquestion of which 2-groups are rea lizable as unit groups of finite\nrings\, a necessary step toward determin ing which nilpotent groups are\nrealizable. This is joint work with Nicho las Werner.\n\nZoom link available 8 hrs before talk. See Eric's recent Jo urnal of Algebra paper (with the same title). This seminar is at an unusua l time because Eric will be speaking from Virginia\, USA.\n LOCATION:https://researchseminars.org/talk/GroupsAndCombinatoricsSeminar/2 / END:VEVENT BEGIN:VEVENT SUMMARY:Dr G. Luke Morgan (FAMNIT\, University of Primorska) DTSTART:20200626T080000Z DTEND:20200626T090000Z DTSTAMP:20260422T225725Z UID:GroupsAndCombinatoricsSeminar/3 DESCRIPTION:Title: Some small progress on the PSV Conjecture\nby Dr G. Luke Morgan (FAMNIT\, University of Primorska) as part of Groups and C ombinatorics Seminar\n\n\nAbstract\nThe subject of the talk is the questio n of bounding the number of automorphisms of arc-transitive graphs in term s of the valency of the graph. More specifically\, we consider the questio n for groups acting arc-transitively on graphs such that the local action (that induced on the neighbours of a vertex by the stabiliser of that vert ex) is semiprimitive. This question was originated by Weiss for the case o f primitive local action and generalised by Praeger for the case of quasip rimitive local action. I will report on some recent small progress on the first type - that of semiprimitive local action. The result is akin to Tut te’s famous result on cubic s-arc transitive graphs where the number of automorphisms is bounded by 3*2^(s-1). Tutte's proof was elegant\, element ary and self-contained. The recent progress relies on some group theoretic al tools that were developed for use in the Classification of the Finite S imple 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 keeping in mind the ``big picture'' concerning where progress now stands on these co njectures.\n LOCATION:https://researchseminars.org/talk/GroupsAndCombinatoricsSeminar/3 / END:VEVENT BEGIN:VEVENT SUMMARY:Gareth Tracey (Renyi Institute\, Budapest) DTSTART:20200703T080000Z DTEND:20200703T090000Z DTSTAMP:20260422T225725Z UID:GroupsAndCombinatoricsSeminar/4 DESCRIPTION:Title: On the Chebotarev invariant of a finite group\nb y Gareth Tracey (Renyi Institute\, Budapest) as part of Groups and Combina torics Seminar\n\n\nAbstract\nGiven a finite group X\, a classical approac h to proving that X is the Galois group of a Galois extension K/Q can be d escribed roughly as follows: (1) prove that Gal(K/Q) is contained in X by using known properties of the extension (for example\, the Galois group of an irreducible polynomial f (x) ∈ Z[x] of degree n embeds into the symm etric group Sym(n))\; (2) try to prove that X = Gal(K/Q) by computing the Frobenius automorphisms modulo successive primes\, which gives conjugacy c lasses in Gal(K/Q)\, and hence in X. If these conjugacy classes can only o ccur in the case Gal(K/Q) = X\, then we are done. The Chebotarev invariant of X can roughly be described as the efficiency of this “algorithm”. In this talk we will define the Chebotarev invariant precisely\, and descr ibe some new results concerning its asymptotic behaviour.\n LOCATION:https://researchseminars.org/talk/GroupsAndCombinatoricsSeminar/4 / END:VEVENT BEGIN:VEVENT SUMMARY:Saul Freedman (University of St Andrews) DTSTART:20200731T080000Z DTEND:20200731T090000Z DTSTAMP:20260422T225725Z UID:GroupsAndCombinatoricsSeminar/5 DESCRIPTION:Title: The non-commuting\, non-generating graph of a group< /a>\nby Saul Freedman (University of St Andrews) as part of Groups and Com binatorics Seminar\n\n\nAbstract\nGiven a group G\, we can construct assoc iated graphs that encode certain relations between the elements (or subgro ups) of G. A well-known example is the generating graph of G\, whose verti ces are the nontrivial elements of G\, with two vertices joined if the ele ments form a generating set for G. In June this year\, Burness\, Guralnick and Harper showed that if the generating graph of a finite group has no i solated vertices\, then it as "dense" as possible\, in the sense that it i s connected with diameter at most 2. This generalises a famous result of B reuer\, Guralnick and Kantor from 2008: the generating graph of a non-abel ian finite simple group is connected with diameter 2.\n\nConsider now the non-commuting\, non-generating graph of G\, obtained by taking the complem ent of the generating graph\, removing edges between elements that commute \, and finally removing vertices corresponding to elements of Z(G). In thi s talk\, we explore the connectedness and diameter of this graph for finit e (and certain infinite) groups G\, for example by studying the maximal su bgroup structure of G. In particular\, we prove a result that is perhaps s urprising: in many cases\, this naturally-defined subgraph of the compleme nt of the dense generating graph is itself similarly dense.\n\nWe also pre sent in this talk a new upper bound on the diameter of a related graph: th e intersection graph of a finite non-abelian simple group. The vertices of this graph are the nontrivial proper subgroups of the group\, with two su bgroups joined if they intersect nontrivially.\n\nPassword hint: 047877+ t he order of Alt(5)\n LOCATION:https://researchseminars.org/talk/GroupsAndCombinatoricsSeminar/5 / END:VEVENT END:VCALENDAR