The Centre for the Mathematics of Symmetry and Computation organises a number of open seminars, public lectures, and events.

​ The Groups and Combinatorics Seminar occurs regularly on Friday afternoons at 16:00. There will be `cake’ in the Mathematics and Statistics tea room at 15:40 beforehand, and after 17:05 we go to the University Club for a drink. Everyone is welcome to join us. Please email the seminar convener Stephen.Glasby@uwa.edu.au if you would like to give a seminar.

2022 Seminars

Time and Place: 4pm Fri 25 March 2022, MS Teams Speaker: Tony Li (University of Western Australia) Title: The chromatic polynomial of a graph and its roots

Time and Place: 4pm Fri 01 April 2022, Weatherburn LT, MS Teams Speaker: Michael Giudici (University of Western Australia) Title: $2$-closed groups and automorphism groups of digraphs Abstract: Wielandt introduced the notion of the 2-closure of a permutation group $G$ on a set $\Omega$. This is the largest subgroup of $\mathrm{Sym}(\Omega)$ with the same set of orbits on ordered pairs as $G$. We say that $G$ is 2-closed if $G$ is equal to its 2-closure. The automorphism group of a graph or digraph is a 2-closed group. In this talk I will discuss some recent work with Luke Morgan and Jin-Xin Zhou on 2-closed groups that are not the automorphism group of a graph or digraph.

Time and Place: 4pm Fri 22 April 2021, Weatherburn LT, MS Teams Speaker: Heiko Dietrich (Monash University) Title: Computing with groups Abstract: Many groups (and other algebraic structures) can be described concisely by small generating sets, which makes them suitable for machine computations. My talk will report on some common ways to describe groups for a computer and how to compute them, and then focus on a survey of known and new quotient algorithms.

The Maths and Stastistics Coloquium on Thursday 28 April will be by Heiko Dietrich.

Time and Place: 4pm Thu 28 April 2022, Blakers LT Speaker: Heiko Dietrich (Monash University) Title: Deciding that two groups are the same: an update on group isomorphism Abstract: Groups are mathematical objects that abstractly capture the concept of symmetry. Two groups are isomorphic if they essentially describe the same data, but they might be given with respect to different frameworks of reference. More specifically, two groups are isomorphic if there is a 1-to-1 correspondence between their elements that preserves the group operations. The group isomorphism problem is a decision problem that asks whether two given groups are isomorphic. This problem is of interest in theoretical computer science (with focus on the complexity of the problem) and in computational group theory (with focus on practical algorithms). My talk will survey some known results and comment on some recent new results, covering both complexity theory and practical applications to computer algebra systems.

Time and Place: 4pm Fri 06 May 2022, Weatherburn LT, MS Teams Speaker: Pablo Spiga (Università degli Studi di Milano-Bicocca) Title: Erdős-Ko-Rado theorems for permutation groups Abstract: The Erdős-Ko-Rado theorem is a classical result in extremal set theory which has remarkable generalizations in various areas of mathematics. In this talk, we are interested in a generalization of the Erdős-Ko-Rado theorem for permutation groups. In this talk, we review some of the early results of Erdős on intersecting sets of permutations and some recent developments.

Time and Place: 4pm Fri 20 May 2022, Weatherburn LT, MS Teams Speaker: Andrea Lucchini (University of Padova) Title: The soluble graph and the Engel graph of a finite group Abstract: In the commuting graph of a finite group G there is an edge between two vertices x and y if <x,y> is abelian. We may generalize this definition in two different directions. Given a class F of finite groups, the F-graph of G is the undirected graph in which there is an edge between x and y if <x,y> is in F. Given a word w in the free group of rank 2, the w-graph of G is the direct graph in which there is an edge x->y if w(x,y)=1. In the first part of the talk we will discuss the connectivity properties of the F-graphs, with particular attention to the case when F is the class of the finite soluble groups, describing some results obtained in a joint paper with Tim Burness and Daniele Nemmi. In the second part of the talk we will consider the n-Engel graph, i.e. the w-graph where w is the n-Engel word and we will define the Engel graph of G as the graph whose vertices are the elements of G which does not belong to the hypercenter of G, and in which there is an edge x->y if the n-Engel word is trivial in x and y, for some positive integer n. In a joint paper with Eloisa Detomi and Daniele Nammi, we proved that if G modulo the hypercenter is neither an almost simple group nor a Frobenius group, then the Engel graph of G is strongly connected. The connectivity properties of the Engel graphs of the almost simple groups are investigated in a very recent joint work with Pablo Spiga.

Time and Place: 4pm Fri 27 May 2022, Weatherburn LT, MS Teams Speaker: Gordon Royle (University of Western Australia) Title: Computational Combinatorics, Binary Matroids and Cubelike Graphs (cancelled) Abstract: In 1976, Appel and Haken announced a proof of the 4-colour theorem that fundamentally relied on extensive computer calculations that could never be verified by hand. Despite the vehement negative reactions to this announcement by various prominent mathematicians and philosophers and their dire warnings about the death of proof, mathematics has somehow survived its predicted collapse. Combinatorics in particular has not only survived, but has grown enormously over the last 50 years, not despite the use of the computers, but at least partly because of the use of computers. Computational combinatorics is an umbrella term describing a style of research that involves “combining pure mathematics, algorithms and computational resources to solve problems in pure combinatorics”. This broad term encompasses many computational tasks, including direct search for examples or counterexamples, construction of databases of combinatorial objects, experimenting with small cases to develop or fine-tune conjectures consistent with the data, and so on. Almost all of my own research could be described as computational combinatorics, but often the final papers downplay (or even omit) details of the computations that were instrumental in finding the results. In this talk, I will describe some of the computational techniques, tools and software that I have used, focusing on my research into binary matroids and cubelike graphs as illustrative examples throughout. No specific prior knowledge of binary matroids or cubelike graphs is needed to follow the talk, because it turns out that both binary matroids and cubelike graphs are essentially just sets of vectors in a binary vector space, just viewed from different perspectives (which will be fully explained). I will be giving this talk at the AMSI Workshop “Bridging Mathematics and Computer Science” in Sydney the following week, so constructive criticism is welcomed.

Time and Place: 4pm Fri 03 June 2022, Robert Street LT, MS Teams Speaker: Stephen Glasby (University of Western Australia) Title: Derangements in wreath products Abstract: A derangement is a permutation with no fixed points. Much research has gone into studying the proportion of derangements in a finite permutation group G. This talk was motivated by the question: What can we say about the proportion of derangements in a large primitive permutation group? Such a group preserves a cartesian power structure and its socle is a power of an alternating group. I will make several connections with past research and some classical combinatorics. This is joint work with V. Arumugam and H. Dietrich.

Time and Place: 4pm Fri 1 July 2022, Weatherburn LT, MS Teams Speaker: Alice Devillers (University of Western Australia) Title: J-groups Abstract: At the 2018 CMSC retreat, Tim Boykett introduced a peculiar group condition in the form of a functional equation: We say a group G is a J-group if there exists an element k (called witness) and a function f from G to G such that f(xk)=xf(x) for all x in G. I will explain what we found out about this property and the set of witnesses, as well as the twists and turns that led from the retreat to the publication of our results in 2022. In particular, we will be interested in the question of which p-groups are J-groups. This is joint work with Dominik Bernhardt, Tim Boykett, Johannes Flake, Stephen Glasby.

Time and Place: 4pm Fri 15 July 2022, Weatherburn LT, MS Teams Speaker: Jesse Lansdown (University of Western Australia) Title: Small Schurian association schemes Abstract: Association schemes describe highly regular combinatorial structures and are important in coding theory, geometry, group theory, and even statistics. An association scheme arising from a transitive group is called Schurian. With the recent classification of transitive permutation groups of degree 48 by Holt, Royle, and Tracey, we are able to classify the Schurian association schemes of order at most 48 and consequently the 2-closed transitive permutation groups of corresponding degree. In this seminar I will present details of the computational aspects of this classification, as well as an overview of some of the properties of the resulting association schemes.

Time and Place: 4pm Fri 22 July 2022, Weatherburn LT, MS Teams Speaker: Lie Chen (University of Western Australia) Title: The maximal subgroups of the finite symplectic groups Abstract: The research of the maximal subgroups of the finite classical groups is intimately connected to the study of the primitive permutation groups. In 1984, Aschbacher developed a fundamental theorem that roughly depicts the subgroups of almost all of the finite almost simple classical groups, which divides these subgroups into nine classes. The result was then enriched by Kleidman and Liebeck in 1989. They provided remarkably detailed enumeration of the maximal subgroups of geometric type of the almost simple finite classical groups of dimension greater than 12. In this talk I will discuss Aschbacher’s theorem in the finite symplectic groups.

Time and Place: 4pm Fri 12 Aug 2022, Weatherburn LT, MS Teams Speaker: Anton Baykalov (University of Western Australia) Title: Intersection of conjugate solvable subgroups in finite groups Abstract: Consider the following problem stated by Vdovin (2010) in the “Kourovka notebook” (Problem 17.41): Let $H$ be a solvable subgroup of a finite group $G$ that has no nontrivial solvable normal subgroups. Do there always exist five conjugates of $H$ whose intersection is trivial? This problem is closely related to a conjecture by Babai, Goodman and Pyber (1997) about an upper bound for the index of a normal solvable subgroup in a finite group. The problem was reduced by Vdovin (2012) to the case when $G$ is an almost simple group. In this talk, we discuss the latest progress on the problem.

Time and Place: 4pm Fri 19 Aug 2022, Weatherburn LT Speaker: Cheryl E. Praeger (University of Western Australia) Title: The pleasure of working with Jan Saxl Abstract: Jan Saxl was Professor of Algebra at Cambridge, and a Fellow of Gonville and Caius College, until his death in 2020. Jan and I were exact contemporaries as DPhil students of Peter Neumann in Oxford. Our first research collaboration began six years later, a year before the finite simple group classification (FSGC) was announced. My talk will reflect on our research collaborations, often also with Martin Liebeck. A major theme was application of the FSGC on problems involving permutation groups, graph symmetry, and simple group structure. I wrote this talk to give at the Jan Saxl Memorial Day which Martin Liebeck and I organised in Cambridge on July 23.

Time and Place: 4pm Fri 02 Sep 2022, Weatherburn LT, MS Teams Speaker: Stephen Glasby (University of Western Australia) Title: An application of the Expander Mixing Lemma. Abstract: This talk was motivated by a key problem in computational group theory, namely recognising classical groups. I shall briefly review techniques in algebraic graph theory before introducing the expander mixing lemma. The EML can be used to establish a very delicate inequality involving a geometric problem needed for classical group recognition. The problem is to find a lower bound for the proportion of pairs of complementary non-degenerate subspaces of prescribed dimensions. Once the eigenvalues are known, the computation is straightforward, but finding the eigenvalues involves some weighty machinery. This is joint work with Ferdinand Ihringer (Ghent) and Sam Mattheus (VUB).

Time and Place: 12-2pm Thu 8, 15, 22, 29 Sep 2022, Zoom Speaker: Alexander A. Ivanov (Imperial College London) Title: Sporadic Groups and Locally Projective Graphs Abstract: In [S. P. Norton, The Monster is fabulous, Contemp. Math., in: Finite Simple Groups: Thirty Years of the Atlas and Beyond, 694, pp.3-10, AMS, Providence, RI, 2017], Simon Norton wrote: “It is often said that Euclid’s ‘Elements’ was intended not so much as an introduction to geometry but to show how to construct the five Platonic Solids; these solids had (in modern terminology) the most elaborate symmetry groups that had been encountered at that time, and were known to be of special interest. On the way to constructing them it would have been necessary to expound most of the geometry that was known at the time. An extrapolation of this statement leads to an uncommon opinion that the most important outcome of the Classification of Finite Simple Group is the discovery of the Sporadic Simple Groups and most importantly the Monster Group. In the foundation course I would like to review the history of the discoveries of the Sporadic Simple Groups along with discussion of the old and new mathematical theories inspired by these discoveries. The theory of Locally Projective Graphs is one of them and will be discussed in detail. Let I = (V,L) be a vertex-line incidence system which is said to be thin or thick if every line contains two or three vertices, respectively. In either case any two lines can intersect in at most one vertex. Let \Gamma be the collinearity graph of I and let G be a group of automorphisms of I. Then \Gamma is said to be locally projective in dimension n with respect to G if (a) G is flag-transitive on I; (b) the stabilizer G(x) of a vertex x \in V in G induces on the set of lines containing x, the group L_n(2) for some n \ge 2 in its natural doubly transitive action of degree 2^n-1. Thus, a locally projective graph in dimension n has valency 2^{n}-1 or 2(2^{n}-1) depending on whether it is thin or thick. The G(x)-invariant structure of a projective GF(2)-space of linear dimension n on the set of lines containing x, will be denoted by \Pi_x. In the thin case we further assume that (c) the action of G on \Gamma is 2- but not 3-arc transitive and (d) the action is of collineation type, meaning that for an edge {x, y} of \Gamma an element in G which swaps x and y can be chosen to centralize G(x) \cap G(y) modulo O_2(G(x) \cap G(y)). The first examples of locally projective graphs come from affine (thin) and projective (thick) GF(2)-spaces. Less trivial classical examples are the dual polar GF(2)-spaces of orthogonal (thin) and symplectic (thick) types. Further examples coming from Petersen and Tilde geometries are associated with some Sporadic Simple Groups: M_{22}, M_{23}, M_{24}, He, Co_2, Co_1, J_4, BM and the Monster Group M. The classification of simply connected Locally Projective Graphs is an ongoing project [A. A. Ivanov, Locally projective graphs and their densely embedded subgraphs, Beitr. Algebra Geom. 62 (2021), 363-374; A characterization of the Mathieu-Conway-Monster series of locally projective graphs, J. Algebra 607A (2022), 426-453].

Time and Place: 4pm Fri 16 Sep 2022, Weatherburn LT, MS Teams Speaker: Nick Gill (Open University, UK) Title: Binary actions of simple groups Abstract: A permutation action of a group G on a set X is “binary” if, roughly speaking, one can deduce the orbits on X^n (for all positive integers n) from the orbits on X^2. This notion arose first in model theory, thanks to the work of Gregory Cherlin. I’m interested in the following question: if G is simple, what are its TRANSITIVE binary actions? Thanks to the work of various authors we know that there are no PRIMITIVE binary actions but, until recently, that was pretty much the extent of our knowledge. It turns out that interesting transitive binary actions do exist for G simple and I will describe some of these. If there is time, I will also describe a nice graph that connects the notion of a transitive binary action to the classical notion of a “strongly embedded subgroup”. This connection offers hope that it may be possible to classify all such actions and I will describe recent progress in this direction.

Time and Place: 4pm Fri 14 Oct 2022, Lecture Room 3 (G02), MS Teams Speaker: Natalia Maslova (RAS and Krasovsky Institute of Mathematics and Mechanics) Title: On characterization of a finite group by its Gruenberg-Kegel graph Abstract: This talk is based on a series of joint works with P.J. Cameron, W. Guo, K.A. Ilenko, A.P. Khramova, A.S. Kondrat’ev, V.V. Panshin, and A.M. Staroletov. The Gruenberg-Kegel graph or the prime graph of a finite group G is the simple graph whose vertices are the prime divisors of the order of G, with primes p and q adjacent in this graph if and only if G contains an element of order pq. Recently P.J. Cameron and the speaker have proved that if there are only finitely many groups whose Gruenberg-Kegel graphs coinside with Gruenberg-Kegel graph of a group G, then G is forced to be almost simple, in particular, any group which is uniquely determided by its Gruenberg-Kegel graph is almost simple. In this talk we discuss the question of characterization of finite simple groups by their Gruenberg-Kegel graphs.

Time and Place: 4pm Fri 28 Oct 2022, Webb LT Geography/Geology Bldg, MS Teams Speaker: Tim Burness (University of Bristol, UK) Title: Fixed point ratios for primitive groups and applications Abstract: Let G be a finite permutation group. The fixed point ratio of an element x in G, denoted fpr(x), is the proportion of points fixed by x. Fixed point ratios for finite primitive groups have been studied for many decades, finding a wide range of applications. In this talk, I will discuss some recent joint work with Bob Guralnick where we determine the triples (G,x,r) such that G is primitive, x has prime order r and fpr(x) > 1/(r+1). This turns out to have some interesting applications and we can use it to obtain new results on the minimal degree and minimal index of primitive groups. Another application arises in joint work with Moreto and Navarro on the commuting probability of p-elements in finite groups.

Time and Place: 4pm Fri 10 Nov 2022, Webb LT Geography/Geology Bldg LT Speaker: Geertrui Van de Voorde (University of Canterbury, NZ) Title: Quasi-quadrics in finite projective spaces Abstract: (joint work with J. Schillewaert) The study of point sets with few intersection numbers is at the core of finite geometry. In 1955, Segre proved a seminal result characterising ovals in Desarguesian planes of odd order as conics; he showed that in those planes, a set of points with the same combinatorial properties as a conic (that is, the same size and the same intersection sizes with lines), must be a conic. In higher dimensions, the same idea leads to the concept of quasi-quadrics; these are a set of points with the same size and intersection numbers with respect to hyperplanes as a classical polar space. Using a technique called pivoting, De Clerck, Hamilton, O’Keefe and Penttila constructed quasi-quadrics that are not quadrics. In this talk, we will introduce quadrics, the pivoting construction, and present some related recent results.

Time and Place: 4pm Fri 18 Nov 2022, Webb LT, MS Teams Speaker: Fredrich Rober (RWTH Aachen University) Title: Black Box Recognition of Wreath Products Abstract: The matrix group recognition project is an international research effort to design and implement efficient algorithms to work with groups of matrices defined over finite fields. The general approach is to decompose a matrix group recursively into its composition factors, i.e. into simple groups, and then to constructively recognise those. A similar approach could be used to decompose a permutation group recursively into primitive groups. But in contrast to simple groups, not much is known about the constructive recognition of primitive groups. In this talk, I will present the current status of my PhD project regarding the constructive recognition of (primitive) wreath products given in black box representation. The wreath products I consider are the ones that can be realised as primitive permutation groups of PA type, e.g. large base primitive groups.

Time and Place: 4pm Fri 25 Nov 2022, Weatherburn LT, MS Teams Speaker: Tim Boykett (Time’s Up; Johannes-Kepler University, Linz; Design Strategies, University of Applied Arts, Vienna) Title: Reversible computation with and without memory Abstract: Algebraic models of computation have been developed in order to express many varied aspects of theories of computation. For instance Clone Theory emerged from the work of Post, Malcev and others, trying to understand the collections of computations that could be represented as gates, mappings from A^n to A for some signal or state set A. Clone theory has a well explored and powerful dual theory of relations. This talk looks at the question of computation as collections of gates from A^n to A^n that are closed under natural operations. While the theory is more general, we are particularly interested in reversible gates, that is, permutations of A^n. Such collections have been studied for instance by Peter Cameron, Ben Fairbairn and Max Gadouleau as computation without memory. Two ways in which computer scientists interested in reversible computation add memory is to talk about borrowed and ancilla bits, extra input-outputs that must be treated in certain ways. Ancilla bits are supplied and must be returned (output) in a particular state; borrowed bits can be supplied in any state and must be returned in that same state. We will talk about generation in reversible gate sets and will share a description of the maximal gate sets without memory. We will outline our understanding of maximal gate sets closed with ancilla and borrowed memory, using Scott Aaronson, Daniel Grier and Luke Schaeffer’s analysis of ancilla closed classes of reversible gates on a state set of order 2. We will introduce Emil Jerabek’s dual structure to memoryless gate sets and will speculate on the version that should be used for ancilla and borrow memory. The talk is necessarily interdisciplinary. It will be of interest to and accessible to researchers with an interest in permutation groups, computational modelling, clone theory, theoretical computer science or related fields.

Time and Place: 4pm Fri 02 Dec 2022, Webb LT, MS Teams Speaker: Dan Hawtin (University of Rijeka, Croatia) Title: Neighbour-transitive codes in Kneser graphs Abstract: A code is a subset of the vertex set of a graph. Classically codes have been studied in the Hamming and Johnson graphs; here we study codes in Kneser graphs. The vertex set of a Kneser graph K(n,k) is the set of all k-subsets of an n-set V. Note that if n=2k+1 then K(n,k) is an odd graph. A code C is s-neighbour-transitive if its automorphism group Aut(C) acts transitively on each of the sets C=C_0, C_1,…, C_s, where {C=C_0, C_1,…, C_\rho} is the distance partition, and \rho is the covering radius, of C. First, we give a full classification of 2-neighbour-transitive codes in Kneser graphs with minimum distance at least 5. We then consider neighbour-transitive (i.e., 1-neighbour-transitive) codes in Kneser graphs. If C is a neighbour-transitive code in a Kneser graph and Aut(C) acts intransitively on V then we characterise C in terms of certain parameters and give several examples. If C is a neighbour-transitive code in an odd graph and C acts imprimitively on V, we again characterise C via certain parameters, giving an example in each case. Finally, we provide a structural result for the case that C is a code in a Kneser graph with n>2k+1 (i.e., not an odd graph), C has minimum distance at least 3, and Aut(C) acts transitively on V.

Upcoming 2022 Seminars (details may change)

Time and Place: 4pm Fri XXX 2022, Weatherburn LT, MS Teams Speaker: XXX Title: Abstract:

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2021 Seminars

Speaker: Stephen Glasby (University of Western Australia) Title: Recognizing simple groups: alternating and classical Time and place: 26 Feb 2021, 4pm at Weatherburn LT Abstract: The structure of a non-solvable finite group is strongly influenced by its non-abelian composition factors. A central problem in computational group theory, therefore, involves quickly recognizing non-abelian simple composition factors. I will consider the most commonly occurring non-abelian simple groups namely the alternating (Alt_n) or the classical groups (PSL_d(q), PSp_d(q), PSU_d(q), POmega_d(q)). The idea is to find by random selection group elements with certain properties including being relatively abundant. This is joint work with Niemeyer, Praeger and Unger.

Speaker: Alice Niemeyer (RWTH Aachen University) Title: Recognising Classical Groups Time and place: 4pm Fri 03 Mar 2021, Weatherburn LT Abstract: Algorithms to facilitate efficient computation with finite groups are a rich source of interesting problems. In this talk we discuss some of the problems that arise when designing an algorithm to find computationally useful generating sets for finite classical groups.

Speaker: Jack Saunders Title: Cohomology of PSL(2,q) Time and place: 4pm Fri 26 Mar 2021, Zoom + Weatherburn LT Abstract: Group cohomology is intricately tied to the extension theory of finite groups and their representations. In this talk, we will give a brief overview of the area along with a generalisation of a result of Guralnick and Tiep from 2011. We will then discuss the cohomology of PSL(2, q) in cross (non-defining) characteristic, along with that of PGL(2, q) and SL(2, q) if time permits.

Speaker: Cheryl Praeger (University of Western Australia) Title: Finite edge-transitive Cayley graphs, quotient graphs and Frattini groups Time and place: 4pm Fri 09 Apr 2021, Weatherburn LT Abstract: The edge-transitivity of a Cayley graph is most easily recognisable if the subgroup of “affine maps” preserving the graph structure is itself edge-transitive. These are the so-called normal edge-transitive Cayley graphs. Each of them determines a set of quotients which are themselves normal edge-transitive Cayley graphs, and which are built from a very restricted family of groups (direct products of simple groups). We address the questions: how much information about the original Cayley graph can we retrieve from this special set of quotients? Can we ever reconstruct the original Cayley graph from them: if so, then how? Our answers to these questions involve a type of “relative Frattini subgroup” determined by the Cayley graph, which has similar properties to the Frattini subgroup of a finite group. I’ll discuss this and give some examples. It raises many new questions about Cayley graphs. This is joint work with Behnam Khosravi.

Speaker: Binzhou Xia (The University of Melbourne) Title: Constructing tetravalent half-arc-transitive graphs Time and place: 4pm Fri 16 Apr 2021, Weatherburn LT Abstract: Half-arc-transitive graphs are a fascinating topic, which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct half-arc-transitive graphs with prescribed vertex stabilizers. In this talk, I’ll focus on the tetravalent case, giving new constructions of half-arc-transitive graphs with various vertex stabilizers. This sheds light on the larger problem of which groups can be the vertex stabilizer of a tetravalent half-arc-transitive graph.

Date: 30 July 2021 Speaker: Sergio Sicchia (TU Kaiserslautern) Location: Weatherburn Street LT Title: Normalizers of simple diagonal groups in polynomial time Abstract: The normalizer problem has as input generating sets X and Y for subgroups G and H of the symmetric group S_n, and asks one to return generators for the normalizer of H in G. In my PhD thesis I developed algorithms for the situation where G is the symmetric group S_n and H is primitive with non-regular socle. These algorithms are both practically efficient and run in polynomial time. In this talk I will focus on primitive groups of simple diagonal (SD) type. I will also present the main tool I use to structure both theory and algorithms: permutation morphisms, a generalization of permutation isomorphisms.

Time and Place: 4pm Fri 06 August 2021, Weatherburn LT Speaker: Jesse Lansdown (University of Western Australia) Title: The existence of synchronising groups of diagonal type Abstract: Motivated originally by synchronisation in automata, there is ongoing work to classify primitive permutation groups within the sychronisation hierarchy. This hierarchy consists of natural classes of groups - synchronising, separating, and spreading - which lie between primitive and 2-transitive. Synchronising primitive groups must be of affine, almost simple, or diagonal type. Until recently, no synchronising groups of diagonal type were known. In this talk I will present recent work (with John Bamberg, Michael Giudici, and Gordon Royle) in which we show that PSL(2, q) × PSL(2, q) acting in its diagonal action on PSL(2,q) is separating, and hence synchronising, for q = 13 and q = 17, and non-spreading for all prime powers of q.

Time and Place: 4pm Fri 13 August 2021, Weatherburn LT Speaker: Kurt Williams (University of Western Australia) Title: Rayleigh-Taylor instability: an application of group theory methods Abstract: The Rayleigh-Taylor instability is a ubiquitous phenomenon of significant interest to both physics and mathematics, occurring whenever two fluids of differing densities are accelerated against the density gradient. The problem is ill-posed, but by careful appeal to the theory of space-groups, a solution can be constructed in full consistency with physics principles. We apply the theory of space groups to resolve the development of structures – both bubbles and spikes – along the interface. By constructing a velocity potential invariant under action of the relevant group, the interface conditions can be reduced to a system of coupled ordinary differential equations, which we solve in the early time “linear” regime and the later time “nonlinear” regime. We construct a geometry-independent form of the nonlinear dynamics including the structure velocity and the interfacial shear. The redimensionalisation transformations are used to compare the relative velocity and shear of both bubble and spike structures and compare the effect of dimensionality and symmetry on Rayleigh-Taylor instability.

Time and Place: 4pm Fri 20 August 2021, Weatherburn LT Speaker: Phill Schulz (University of Western Australia) Title: The automorphism group of an abelian p-group Abstract: This is an expository talk about the structure of Aut(G) for an abelian p-group G. I present a sequence of increasingly complex examples: cyclic, finite, bounded, direct sum of cyclics, simply presented, leading to the open general case. No previous knowledge of abelian group theory is required.

Time and Place: 4pm Fri 27 August 2021, Weatherburn LT Speaker: Ted Dobson (FAMNIT, University of Primorska) Title: A new family of transitive permutation groups,and the solution to two problems on vertex-transitive graphs Abstract: Over the course of my career, I have noticed that many results on symmetries in graphs, especially concerning the Cayley isomorphism problem, determining automorphism groups of vertex-transitive graphs, and also determining minimal transitive subgroups of the automorphism groups of vertex-transitive graphs (essentially classifying them), do not need too much information about the graph being considered. That is, a certain group theoretic property that the automorphism group of a vertex-transitive graph has is all that is needed to prove many results. To capture this idea, we introduce a new class of transitive permutation groups which has this property, and contains all transitive 2-closed permutation groups. There are 3-closed groups which do not have this property, so in a way this class is “between” 2-closed and 3-closed. So we call this class 5/2-closed. It turns out that not only are 2-closed groups contained in this new class, but also the automorphism groups of configurations, and in more generality, combinatorial objects whose “edge” set is a 1-intersecting set system (that is, any two “edges” intersect in exactly one vertex). To demonstrate this idea, we solve two problems in the more general context of 5/2-closed transitive group, and so obtain as corollaries the solutions to the two problems for 2-closed groups. Let p, q, and r, be distinct primes. We show that any 5/2-closed group with a normally (or genuinely) 3-step imprimitive subgroup has a minimal transitive subgroup in one of four families. It is known that for each of these four families, there is a vertex-transitive graph whose automorphism group contains a minimal transitive subgroup in the family but does not have a minimal transitive subgroup in the other three families. Next, we determine all 5/2-closed permutation groups of odd prime-power degree that contain a regular cyclic subgroup, and so also determine all 2-closed permutation groups of odd prime-power order that contain a regular cyclic subgroup.

Time and Place: 4pm Fri 03 September 2021, Weatherburn LT Speaker: Primož Potočnik (University of Ljubljana, Faculty of mathematics and physics) Title: Fixity of vertex-transitive graphs Abstract: Given a connected finite graph Gamma on n vertices, one can ask what is the maximum number of vertices that a nontrivial automorphism of Gamma can fix. This parameter will be called the {\em fixity} of the graph Gamma. The notion of {\em fixity} (or rather its ``opposite’’ notion, the minimum number of points moved by an element of the group, called called {\em minimal degree}) has been studied extensively in the context of permutation groups. For example, a classical result of Jordan states that the symmetric group S_n is the only primitive permutation groups of degree n and fixity n-2. On the other hand, not much is known about the graph theoretical version of the problem. Clearly the fixity of Gamma is maximal possible (that is, n-2) if and only if there exists two vertices that, after removing an edge between them if they are adjacent, have the same neighbourhood, and it is minimum possible (that is, 0), if the automorphism group acts semiregularly on the vertices. However, I will focus on the question what can be said about connected vertex-transitive graphs of small valence (3 and 4, in particular) whose fixity is large. In particular, I will present a solution of a problem that was raised a few years ago, asking for the classification of connected cubic vertex-transitive graphs with fixity larger than 1/3. This solution is a result of joint work with Pablo Spiga.

Time and Place: 4pm Fri 17 September 2021, Weatherburn LT Speaker: Lie Chen (UWA in China) Title: Vertex-primitive s-arc-transitive digraphs of almost simple groups Abstract: The property of s-arc-transitivity has been well studied for many years. Weiss proved that finite undirected graphs that are not cycles can be at most 7-arc-transitive. On the other hand, Praeger showed that for each s there are infinitely many finite s-arc-transitive digraphs that are not (s+1)-arc-transitive. However, G-vertex-primitive (G,s)-arc-transitive digraphs for large s seem rare. Thus we are interested in finding an upper bound for such s. In 2018, Giudici and Xia showed that it is sufficient to determine s when G is almost simple. We will show that s\leq 1 when G is almost simple with socle Sz(2^{2n+1}) or ^{2}G_{2}(3^{2n+1}) for n\geq 1.

Time and Place: 4pm Fri 24 September 2021, Weatherburn LT Speaker: Stephen Glasby (University of Western Australia) Title: On the maximum of the weighted binomial sum $2^{-r}\sum_{i=0}^r\binom{m}{i}$ Abstract: Add the first r+1 binomial coefficients in the row m of Yang Hui’s (aka Pascal’s) triangle and divide by 2^r. This gives a function which appears in probability, coding theory, information theory, and even permutation group theory. For most values of r there is no closed formula for this sum. We prove that the maximum of this function occurs (with 4 exceptions) when r equals Floor(m/3)+1. The proof of this splits into two parts: one is easy and the other is quite difficult requiring an increasing amount of precision. The talk is not technical, and it mentions generalizations and open questions. This is joint with G.R. Paseman.

Event: 4pm Fri 01 October 2021, WIMSIG no seminar

Time and Place: 4pm Fri 08 October 2021, Weatherburn LT Speaker: Natalia Maslova (RAS and Krasovsky Institute of Mathematics and Mechanics) Title: On characterization of a finite group by its Gruenberg-Kegel graph Abstract: The Gruenberg-Kegel graph GK(G) associated with a finite group G is an undirected graph without loops and multiple edges whose vertices are the prime divisors of the order of G and in which vertices p and q are adjacent in GK(G) if and only if G contains an element of order pq. This graph has been the subject of much recent interest. In this talk we discuss a recent progress in characterization of a finite group by its Gruenberg-Kegel graph.

No Seminar: 15 and 22 October 2021

Time and Place: 4pm Fri 29 October 2021, Robert Street LT Speaker: John Bamberg (University of Western Australia) Title: On Shult and Thas’ special sets Abstract: Shult and Thas (“Constructions of polygons from buildings”, Proc. LMS, 1995) showed that a certain configuration of totally singular lines of a nonsingular elliptic quadric in the finite projective space PG(5,q) gives rise to a generalised quadrangle of order (q^2,q^2). This geometric object can also be seen as a certain configuration of totally singular points of the Hermitian space H(3,q^2), and is often called a “special set” in the literature. The generalised quadrangle Q(4,q^2) arises this way, from the classical special set, and it is an open problem to find non-classical special sets. This talk is on joint work with Giusy Monzillo and Alessandro Siciliano, where we characterise special sets from an algebraic combinatorial point of view.

Event: 5th Australian Algebra Conference Wed Nov 17 - Fri Nov 19, 2021. No G&C seminar.

Time and Place: 4pm Fri 26 November 2021, Weatherburn LT Speaker: Dan Hawtin (University of Rijeka)

Title: Neighbour-transitive codes in generalised quadrangles Abstract: A code C in an arbitrary graph \varGamma is a subset of the vertex set of \varGamma. The minimum distance \delta of a code C is the smallest distance between a pair of distinct elements of C and the graph metric gives rise to a partition {C_0,C_1,\ldots,C_\rho} of the vertices of \varGamma where C_0=C and \rho is the maximum distance between any vertex of \varGamma and its nearest element in C. In this talk we consider the case where \varGamma is the point-line incidence graph of a generalised quadrangle {\mathcal Q} and we say that C is a code in the generalised quadrangle {\mathcal Q}. Since the diameter of \varGamma is 4, both \rho and \delta are at most 4. If \delta=4 then C is a partial ovoid or partial spread of {\mathcal Q}, and if, additionally, \rho=2 then C is an ovoid or a spread. A code C in {\mathcal Q} is neighbour-transitive if its automorphism group acts transitively on each of the sets C and C_1. The talk includes some examples, some partial classification results and some open questions.