Title: The p-part of the order of an almost simple group of Lie type

Time and place: 4pm Friday 23 Nov 2018, Robert Street LT (not Weatherburn LT)

Abstract: Primitive permutation groups are fundamental building blocks in the sense that every finite permutation group can be built from the primitive ones. Apart from the alternating group A_n and the symmetric group S_n of degree n, the primitive subgroups G of S_n are small. For example, in 1980 Praeger and Saxl showed that |G| e 4^n, which is much smaller than n!/2. Since this time, powerful results such as the O’Nan-Scott Theorem, which classifies the primitive permutation groups, and the Classification of the Finite Simple Groups, have become available. We will bound the p-part |G|_p of |G| for some prime p. This is, the largest p-power p^{ u_p(G)} that divides |G|. The bound |G| e 4^n implies nu_p(G) e n og_p(4). We prove the stronger bound nu_p(G) e frac{2sqrt{n}}{(p-1)}+1 (with five exceptions). For several cases, we even obtain a bound that is logarithmic in n. Our proof uses the O'Nan-Scott theorem to reduce to simple groups. The hardest case, and the one I will discuss, is when the simple group is of Lie type.

Title: Erdös-Ko-Rado Problem for Permutation Groups

Time and place: 4pm Friday 01 Feb 2019, Weatherburn LT

Abstract: A classical result of Erdös-Ko-Rado in extremal set theory is about intersections of subsets of a set, leading to the so-called Erdös-Ko-Rado problem in various versions. I will explain the problem for permutation group version, and then address a conjecture about the upper-bound for the numbers of intersecting sets.

Title: Conjugacy class sizes in finite groups: variations on the theme

Time and place: 4pm Tuesday 12 Feb 2019, Weatherburn LT

Abstract: The study of conjugacy class sizes goes back to the beginning of the last century with Burnside's result, Miller's studies on groups with few conjugacy classes and Ito who began the study of the structure of a group in terms of the number of conjugacy classes. Over the last 30 years the subject has become fashionable and many papers have been written on this topic. In this talk I try to summarize some results obtained by using two particular graphs: the common divisor graph and the prime graph in relationship with conjugacy class sizes and finally the variation regarding the so-called vanishing classes, that are those classes of elements g for which there exists an irreducible non linear character chi such that chi(g)=0.

Title: An update on the Polycirculant Conjecture

Time and place: 4pm Friday 15 Feb 2019, Robert Street LT

Abstract: One version of the Polycirculant Conjecture is that every finite vertex-transitive digraphs admits a non-trivial semiregular automorphism. I will give an overview of the status of this conjecture, as well as describe some recent progress with Michael Giudici.

Title: Searching for partial congruence partitions in groups of order p^8

Time and Place: 4pm Friday 08 Mar 2019, Weatherburn LT

Abstract: A partial congruence partition (or PCP) of a group is defined simply as a set of pairwise disjoint subgroups which pairwise factorise the whole group. This project has focused on finding examples of PCP that are 'large' in the sense that they are close the best known theoretical bounds on the maximum number of subgroups comprising a PCP. In particular, we focused on the non-elementary abelian groups of order 2^8 and 3^8, where no large examples were previously known. These groups, with one exception, were exhaustively searched for large PCP. In this talk I will discuss some of the theoretical bounds with a focus on deriving results that are useful computationally, before moving on to some details of the computational enumeration.

Title: Elements with large irreducible submodules contained in maximal subgroups of the general linear group

Time and place: 4pm Tuesday 12 Mar 2019, Blakers LT

Abstract: We refer to an element of the finite general linear group GL(V) as being fat if it leaves invariant, and acts irreducibly on, a subspace of dimension greater than dim(V)/2. Fat elements generalise the concept of ppd-elements, which are defined by the property of having orders divisible by certain primes called primitive prime divisors. In 1997, Guralnick, Penttila, Praeger and Saxl classified all subgroups of GL(V) containing ppd-elements. Their work has had a wide variety of applications in computational group theory, number theory, permutation group theory, and geometry. Our overall goal is to carry out an analogous classification of all subgroups of GL(V) containing fat elements.

During my PhD candidature I examined the occurrence of fat elements in GL(V) and various of its maximal subgroups. I showed that, often, this problem can be handled in a uniform way by considering "extremely fat" elements and counting certain irreducible polynomials. In my talk, I will present this method for groups belonging to Aschbacher's C2 class. The results we obtain significantly differ from the findings of the ppd-classification.

Title: Arc-transitive bicirculants

Time and place: 4pm Friday 12 Apr 2019, Weatherburn LT

Abstract: A graph on 2n vertices is a bicirculant if it admits an automorphism that is a permutation with two cycles of length n. For example, the Petersen and Heawood graphs. Arc-transitive bicirculants of valencies three, four and five have previously been classified by various authors. In this talk I will discuss recent joint work with Alice Devillers and Wei Jin that characterises all arc-transitive bicirculants and provides a framework for their complete classification.

Title: Girth, words and diameters of Cayley graphs

Time and place: 4pm Wednesday 17 Apr 2019, Woolnough LT

Abstract: The girth of a graph is the minimal length of a cycle in the graph. Finding regular graphs with large girth relative to their diameter is the subject of much interest, and a fruitful source of examples has been found in Cayley graphs of various families of finite classical groups. I shall discuss some new results in this area on the girth of Cayley graphs of finite classical groups G on random sets of generators. The main tool is a new bound on the probability that a given word w takes the value 1 when evaluated in G, in terms of the length of w.

Title: The 2-transitive permutation representation of the small Ree groups

Time and place: 4pm Friday 03 May 2019, Weatherburn LT

Abstract: Given a group G, the question of which subsets S of G generate G is of natural interest. One approach to this question is determining the Möbius function of G, introduced by Hall in 1936. In order to determine the Möbius function, it is necessary to have a good understanding of the subgroup structure of G, which is of course of interest in its own right. In this talk we discuss how the Möbius function is determined in practice, using the specific case of the small Ree groups.

For upcoming seminars see https://staffhome.ecm.uwa.edu.au/~00059629/GroupsAndCombinatoricsSeminar/S19.html

Title: From Lehman Matrices To (Im)Perfect Graphs

Time and place: 4pm Friday 10 May 2019, Weatherburn LT

Abstract: A pair (A,B) of square 0/1 matrices is called a Lehman pair if AB^T = J + k I where J is the all-ones matrix, I is the identity matrix and k is a positive integer, and an individual square 0/1 matrix is called a Lehman matrix if it belongs to a Lehman pair. The study of such matrices arose independently in the work of Lehman on problems in operations research, and the work of Bridges and Ryser who viewed them as generalisations of certain combinatorial designs. A number of authors have given methods of constructing Lehman matrices, including several recursive constructions that generate larger Lehman matrices from smaller ones, but always with the same value of k. In joint work, Dillon Mayhew, Irene Pivotto and I discovered a curious construction that transforms certain Lehman matrices with k=1 into “Lehman-like” matrices with k=-1 (and vice versa). Although barely mentioned in the literature on Lehman matrices, solutions to the matrix equation AB^T = J - I are essentially equivalent to a class of graphs known as "partitionable graphs", which were the central object of study in the decades-long effort to prove Berge’s Strong Perfect Graph Conjecture by a direct characterisation of minimal imperfect graphs.

In this talk, I will introduce all the necessary background concepts, and describe how such an innocuous definition leads quite naturally to such disparate areas of combinatorics.

Title: Limited geodesic transitivity for finite regular graphs

Time and place: 4pm Friday 17 May 2019, Weatherburn LT

Abstract: Joint work with Wei Jin.

For vertex transitive graphs, transitivity on t-arcs, t-geodesics, or distance t vertex pairs, for t leq s, all give symmetry measures of the graph in balls of radius s about a vertex. If the graph has girth g, and s leq g/2, then the sets of t-arcs and t-geodesics are the same for each t leq s, and so the conditions of s-arc transitivity and s-geodesic transitivity are equivalent. The next cases where s= (g+1)/2 and s=(g+2)/2 are interesting. There are s-geodesic transitive examples that are not s-arc transitive. Those which have s=2 and g=3 are collinearity graphs of point-line incidence geometries. However there is no nice general description for the cases where s= 3 and g is 4 or 5. Our approach has required us to classify, as a bye product, all 2-arc transitive strongly regular graphs, and to examine their normal covers. We have lots to describe, as well as open problems to pose.

Title: Words, permutations, and the nonsolvable length of a finite group

Time and place: 4pm Friday 24 May 2019, Weatherburn LT

Abstract: In group theory, the term “word” denotes any concatenation of variables and their formal inverses, such as xxyx^{-1}zy^{-1}. Words are to group theorists what polynomials are to ring theorists: formal expressions into which elements from a concrete structure (a ring resp. group) can be substituted and which can be used to formulate equations over those structures. One type of question commonly studied in this context is the following: Given a word w(X_1,...,X_d) and a number rho in (0, 1], what can one say about finite groups G in which for some g in G, the equation w(X_1,...,X_d) = g has at least rho|G|^d solutions (g_1,...,g_d) in G^d? In this talk, I will discuss recent results of this form which were achieved in collaboration with Aner Shalev from the Hebrew University of Jerusalem.

Title: Actions of Aut(V) on linearly independent subsets of a vector space V

Time and place: 4pm Friday 02 Aug 2019, Weatherburn LT

Abstract: Let V be a finite dimensional rational vector space. The action of the general linear group Aut(V) on the set of bases of V is the main content of First Year Linear Algebra, so you might think there can be nothing new to say about it; but the subject still holds a few surprises.

For example, if G is a subgroup of V, the stabilizer of G in Aut(V) acts on the set of maximal linearly independent subsets of V contained in G. The orbits of this action determine the indecomposable decompositions of G.

Furthermore, the stabilizer of the set of subgroups of V containing G as a subgroup of finite index acts transitively on a larger set of linearly independent subsets, and this action determines a coarse structure theorem for G.

Title: On transitive automorphism groups of 2-designs

Time and place: 4pm Friday 23 Aug 2019, Weatherburn LT

Abstract: A 2-design with parameters (v,k,lambda) is an incidence structure consisting of a set of v points and a set of b blocks with the incidence relation such that every block is incident with exactly k points, and every pair of points is incident with exactly lambda; blocks. An automorphism group of a 2-design is a group of permutations on points of the design which maps blocks to blocks and preserves the incidence and non-incidence. The main part of this talk is devoted to giving a survey on recent study of 2-designs admitting a flag-transitive automorphism group. I also present some recent results on block-transitive automorphism groups of 2-designs.

Title: Edge-primitive 3-arc-transitive graphs

Time and place: 4pm Friday 30 Aug 2019, Weatherburn LT

Abstract: Let Gamma be a finite simple graph with G = Aut(Gamma). We say that Gamma is edge-primitive if G acts primitively on the edges of Gamma. An s-arc in Gamma is an ordered path of length s. We say that Gamma is s-arc-transitive if G is transitive on the set of s-arcs of Gamma.

In 1981, Weiss proved that there exists no finite s-arc-transitive graph of valency at least 3 for s geq 8. Since then, there has been considerable effort to characterise s-arc-transitive graphs for s leq 7. One interesting family of graphs is that of edge-primitive graphs. Many famous graphs are edge-primitive, such as the Heawood graph and the Higman-Sims graph. In 2011, Li and Zhang classified finite edge-primitive s-arc-transitive graphs for s geq 4. We study the problem of classifying finite edge-primitive 3-arc-transitive graphs. This is joint work with Michael Giudici.

Title: Automorphism groups of simple graphs with few vertex-orbits

Time and place: 4pm Friday 13 Sep 2019, Weatherburn LT

Abstract:In this talk we will investigate how to construct automorphism groups of graphs with few vertex orbits. The base case is to construct automorphism groups with 2 vertex orbits. We will see how to describe their group theoretic structure and give hints towards an algorithm for constructing these groups and towards generalizations to more orbits.

If time permits, we will see how to count isomorphism classes of graphs with the same automorphism group using the table of marks.

Title: Automorphism orbits and element orders in finite groups

Time and place: 4pm Friday 27 Sep 2019, Weatherburn LT

Abstract: Joint with Michael Giudici and Cheryl E. Praeger.

In contrast to other kinds of structures (such as graphs), for groups G, the assumption that the automorphism group Aut(G) acts transitively on G is not interesting to study, as only the trivial group satisfies it. Various weakenings of this condition have been proposed and studied, though. For example, in a paper from 1992, Zhang extensively studied finite groups G with the property that for every element order o in G, the action of Aut(G) on order o elements in G is transitive. He called such finite groups AT-groups. Zhang’s ideas and methods also spurred some interest in the graph-theoretic community, due to a connection with CI-groups (groups G such that any two isomorphic Cayley graphs over G are “naturally isomorphic” via an automorphism of G).

In this talk, we present results on finite groups G that are “close to being AT- groups”, essentially showing that such groups are “almost soluble” (i.e., they have a soluble normal subgroup of bounded index). A finite group G is an AT-group if and only if the numbers of Aut(G)-orbits on G and of distinct element orders in G respectively are equal. Hence we measure the “closeness of G to being an AT-group” by comparing those two numbers, considering both their difference and quotient. Along the way, we obtain a curious quantitative characterisation of the Fischer-Griess Monster group M.

Title: Introduction to the GAP method selection for the working mathematician: Methods, Operations and Filters

Time and place: 4pm Friday 11 Oct 2019, Weatherburn LT

Abstract: GAP object can learn information about themselves that are stored as attributes and are present throughout the whole GAP session. Building on this, GAP has a method selection that dynamically changes applied algorithms depending on newly learned properties of the group. We will explore this on the example of computing the average order of elements in a collection and a group.

Title: On Gruenberg-Kegel graphs of finite groups

Time and place: 4pm Friday 18 Oct 2019, Weatherburn LT

Abstract: Let G be a finite group. The spectrum of G is the set of all its element orders. The Gruenberg-Kegel graph of G is a graph whose vertex set is the set of all prime divisors of the order of G, and two distinct vertices are adjacent in this graph if and only if their product is an element order of G. The spectrum is a very important invariant of a finite group. For example, many finite simple groups are defined by their spectra up to isomorphism. The concept of the Gruenberg-Kegel graph of a finite group widely generalizes the concept of the spectrum. In this talk we discuss some characterizations of finite groups by properties of their spectra and Gruenberg-Kegel graphs.

Title: On pronormality of subgroups of odd index in finite groups

Time and place: 4pm Friday 25 Oct 2019, Weatherburn LT

Abstract: A subgroup H of a group G is pronormal in G if for any element g from G, subgroups H and H^g are conjugate in the subgroup generated by H and H^g. Some problems in finite group theory, combinatorics, and permutation group theory were solved in terms of pronormality (see, for example, results by L. Babai, P. Palfy, Ch. Praeger, and others). Thus, the question of description of pronormal subgroups in finite groups is of interest. Well-known examples of pronormal subgroups in finite groups are normal subgroups, maximal subgroups, Sylow subgroups, Carter subgroups, Hall subgroups of solvable groups, and so on. An important problem is to describe families of pronormal subgroups in finite simple groups.

In 2012, E. Vdovin and D. Revin proved that the Hall subgroups are pronormal in finite simple groups and conjectured that the subgroups of odd index are pronormal in finite simple groups. This conjecture was disproved by A. Kondrat'ev, the speaker, and D. Revin in 2016. However, in many finite simple groups the subgroups of odd index are pronormal. In this talk we discuss a recent progress in the classification of finite simple groups in which the subgroups of odd index are pronormal and some connected questions.

Title: Subdegrees of primitive permutation groups

Time and place: 4pm Friday 01 Nov 2019, Robert Street LT

Abstract: A subdegree of a permutation group is the length of an orbit of a point stabiliser. The study of subdegrees of primitive permutation groups has a long history and has attracted the attention of many researchers. In this talk I will survey some results in this area and discuss some recent work on small subdegrees, constant subdgrees and coprime subdegrees. I will also discuss some applications to graph theory.

Cheese and wine to follow in the Maths common room.

Abstract: We all have our doubts off and on if life is really so wonderful. But that is not what I want to address here. Watching the Jimmy Stewart movie with this title, there was one scene which captured my imagination: the Guardian Angel shows George Bailey how the world would have been without him. Personally, I never had much need to know how the world would have looked without me. However, all other things equal, how would life have been if I had lived in a different time and place, would be something of interest to me! This is the stuff of movies and fairy tales. But at least it is possible to play this as an intellectual game. I was born and raised in Germany before WW II. After getting my Ph.D. in 1962, I married a fellow mathematician and we immigrated to the US one year later, where we taught at a university until our retirements, first at Ohio State and then at Binghamton University. What would life have been if I stayed in Germany, did not get married, were born fifty or one hundred years earlier, or were born in another country? Looking at actual and potential role models over the centuries helped me answer some of these questions. In essence, it got me back to the roots of what shaped my life.