Seminars co-ordinated by the Groups and Combinatorics research group.

On the derangement graph of PGL(2,q) acting on the projective line.

• 12 noon, 3rd November, Maths Lecture Room 1
• Pablo Spiga (UWA)

Given a finite set X and a family S of k-subsets of X, we say that S is intersecting if any two elements in S have non-empty intersection. A classical  theorem in extremal combinatorics of Erdos-Ko-Rado classifies the independent sets of maximal size for  2k<|X|. There are many applications  and generalizations  of this theorem in different areas of mathematics. The extension we are interested in deals with permutation groups. In particular, inspired by a recent paper of Godsil-Meagher, we prove an analogue of the Erdos-Ko-Rado theorem for the derangement graph of PGL(2,q) acting on the projective line.

---

On parabolic triple factorisations of general linear groups.

• 1pm, 27th October, Maths Lecture Room 1
• S. Hassan Alavi

For a group G, an expression G = ABA, where A, B are subgroups of G, is called a triple factorisation of G and is denoted by T = (G, A,B). Such factorisations play a fundamental role as `Bruhat decompositions' in the theory of Lie type groups, and more generally in the study of groups with a (B,N)-pair. Therefore, the triple factorisations of G = GL(n, K) where K is a finite field are of our interest. In this talk we present some results about triple factorisations T = (G, A,B) of G = GL(n, K) where K is a finite field and both A and B are parabolic subgroups of G.

---

Vertex primitive 2-path transitive graphs.

• 1pm, 6th October, Maths Lecture Room 1
• Hua Zhang

Let (a,b,g) be an 2 arc. Then we also have another 2 arc (g, b, a). By identifying these two we get a 2-path [a, b g]. In this talk I will present the work that we have done so far on 2-path transitive graphs, including a general study of such graphs, the structure of the point stabilizers, the primitive type of vertex primitive 2-path transitive graphs, a classification for almost simple type, and a construction of some new half transitive graphs.

---

Involutions in Automorphism Groups.

• 22nd September 2009, Maths Lecture Room 1
• Phill Schultz

Let M be a module over a unital ring, and let M = A + B be a decomposition of M as a direct sum. Then Aut(M) contains the involution which is the identity on A and the map b maps to -b on B.

I identify a class of modules for which this mapping from decompositions to involutions is a bijection. For this class, I use properties of involutions to answer several problems in Module Theory; for example:

Which modules decompose as a direct sum of indecomposables? For which modules is such a decomposition unique up to isomorphism?

Kaplansky's Test Problems:

If N is isomorphic to a summand of M, and M to a summand of N, are N and M isomorphic? If M + M is isomorphic to N + N, is M isomorphic to N? If M + A is isomorphic to N + A, is M isomorphic to N?

---

Tits' buildings and applications to the classification of finite simple groups.

• 15th September 2009, Maths Lecture Room 1
• Alice Devillers
  

For a field K, the group PSL(n+1,K) acts on the projective space PG(n,K). For K=C or R, that group is a simple Lie group. Jacques Tits, Belgian mathematician, invented a combinatorial structure, generalising projective spaces, on which the semisimple Lie groups act naturally. These combinatorial objects can be seen as simplicial complexes or as chamber systems and are called buildings. Tits' Lemma states that for a group acting transitively on a chamber system, the chamber system is simply connected if and only if the group can be `nicely presented' as an amalgam. Since buildings are simply connected, this provides presentations of semisimple Lie groups. There are other groups acting transitively on some subchamber systems of buildings. With Bernhard Muhlherr, we provided a local combinatorial criterion for a chamber system to be simply connected. This criterion is then used by Bennett, Gramlich, Hoffman, Shpectorov to get nice presentations of these groups, called Curtis-Tits-Phan presentations. Most finite simple groups (in particular the classical ones) have an action on buildings or on subchamber systems of buildings. Hence we get nice amalgam presentations for them. This allows a `local recognition' of these groups and is used in the ongoing reproving of the classification of finite simple groups (Gorenstein, Lyons and Solomon).

This talk will be an extended version of the talk I will give in Adelaide for AustMS. It is also related to the subject of our new Buildings, finite geometries and Phan-systems brain-storming sessions.

---

Involutions in Classical Groups.

• 8th September 2009, Maths Lecture Room 1
• Michael Giudici
  

Involutions played a key role in the Classification of Finite Simple Groups and many problems concerning simple groups often rely on studying properties of involutions. In this talk I will give a gentle introduction to involutions in classical groups of odd characteristic. This will include their conjugacy classes, representatives and how they act on the natural module.

---

An introduction to GAP.

• 1st September 2009, Maths Lecture Room 1.
• John Bamberg
  

GAP (Groups, Algorithms and Programming) is the most popular open-source computer algebra system that deals with groups, but it is becoming increasingly valuable in working with graphs, designs, semigroups, crystallographic groups, Lie algebras and finite geometries. In this talk, I will give a basic introduction to GAP, starting with the basics of how it works and its structure.

---

Proportions of pre-involutions in finite classical groups.

• 25th August 2009, Maths Lecture Room 1
• Tomasz Popiel
  

Several recent constructive recognition algorithms for finite classical groups are based on finding, by random selection, an element that powers up to an involution in a specified conjugacy class. We present some new results regarding proportions of such elements.

---

Regular and chiral polytopes

• 19 June 2009, Blakers Lecture Theatre
• Alice Devillers
  

Abstract polytopes were introduced by Grunbaum in the 70s. They only carry the combinatorial properties of the face lattice of convex polytopes, without being embedded in a Euclidean space. Regular polytopes are the polytopes with maximal symmetry. Chiral polytopes are the non-regular polytopes with maximal rotational symmetry. I will give an introduction to the subject, as well as some recent results on finite chiral polytopes.

---

Basic incidence geometries

• 12 June 2009, Blakers Lecture Theatre
• Geoffrey Pearce
  

An incidence geometry consists of a set of points of different types, some of which are pairwise incident with one another (subject to certain special conditions). In this talk I will discuss some recent research on 'basic' incidence geometries - these are the fundamental structures which arise when studying incidence geometries by taking normal quotients. Similar applications of normal quotients have proved very successful in the study of other combinatorial structures, such as 2-arc transitive graphs, and take advantage (as they do in the incidence geometry context) of Praeger's characterisation of quasi-primitive permutation groups.

---

Symmetric graphs of diameter two arising from subgroups of the general linear group

• 5 June 2009, Blakers Lecture Theatre
• Carmen Amarra

Let V be a vector space of dimension d over a field of q elements, and let $H \leq GL(V)$. Let S be an orbit of H on V \{0} with -S = S, and define $\Gamma$ to be the graph with vertex set V and whose edges are the pairs {v,w} where $v-w \in S$. Then $\Gamma$ is G-arc-transitive, where $G = V \rtimes H$. It has diameter 2 if V \ S is contained in S+S. We organise our investigation using Aschbacher's Theorem, which identifies eight families of subgroups of the general linear group, and we begin the talk with a description of these. In the next part we present the symmetric diameter 2 graphs that arise in the case where H is a maximal subgroup (as defined in the theorem), for some of these families.

---

The Burnside Problem

• 29 May, 2009, Blakers Lecture Theatre
• Daniel Harvey (University of Adelaide)

In 1902, William Burnside wrote “A still undecided point in the theory of discontinuous groups is whether the order of a group may not be finite, while the order of every operation it contains is finite”. In modern terms, we ask whether a finitely generated group of finite exponent is necessarily finite. In this talk, we shall explore what this means, before considering the problem for cases of small exponent and more generally.

---

Generalised hexagons

• 8 May, 2009, Blakers Lecture Theatre
• John Bamberg (UWA)

I will begin with an overview of generalised polygons, a central part of modern finite geometry, and in particular the known families of generalised hexagons. Towards the end of the talk, I will present some recent work with Nicola Durante (University of Naples) on Split Cayley hexagons. No prior advanced knowledge of finite geometry will be assumed and the main aim of the talk is to provide an introduction to this area of research.

---

Symmetric graphs of diameter two arising from subgroups of the
general linear group

• 5 May, 2009, Blakers Lecture Theatre
• Carmen Amarra (UWA)

Let V be a vector space of dimension d over a field of q elements, and let $H \leq GL(V)$. Let S be an orbit of H on V\{0} with -S = S, and define $\Gamma$ to be the graph with vertex set V and whose edges are the pairs {v,w} where $v-w \in S$. Then $\Gamma$ is G-arc-transitive, where $G = V \rtimes H$. It has diameter 2 if V\S is contained in S+S. We organise our investigation using Aschbacher's Theorem, which identifies eight families of subgroups of the general linear group, and we begin the talk with a description of these. In the next part we present the symmetric diameter 2 graphs that arise in the case where H is a maximal subgroup (as defined in the theorem), for some of these families.

---

Neighbour Transitive Codes

• 1 May, 2009, Blakers Lecture Theatre
• Neil Gillespie (UWA)

Error correcting codes are used to correct errors which occur during the transmission of a message through a noisy communication channel. An underlying principle that is often assumed is that each individual error has an equal probability of occurring during transmission. In this talk we examine this principle from a group theoretic perspective. We embed our codes in a Hamming graph and consider them as subsets of the vertex set. In order to examine the principle that each error is equally likely we introduce the notion of a neighbour transitive code. We shall demonstrate some preliminary results relating to neighbour transitive codes. We also introduce an infinite family of codes with the property that each code is not fixed setwise by the `neighbour' stabiliser. We also investigate the case where the code is fixed setwise by the 'neighbour' stabiliser. In particular we analyse the group structure of the group associated with the code.

---

Finite Bruck loops

• 3 April, 2009, Blakers Lecture Theatre
• Alexander Stein (Freie Universität Berlin)

A Bruck loop is a loop X, in which any elements x, y, z satisfy the (right) Bol identity ((xy)z)y = x ((yz)y) and the map $\iota: X \to X, x \mapsto x^ {-1} $ is an automorphism of the loop $X$. One can think of Bruck loops as generalised (nonassociative) abelian groups. However there exists a simple nonabelian Bruck loop of size 96, in which $\iota$ is the identity (Nagy, Baumeister+Stein, 2007). G.Nagy found even an infinite family of finite simple nonabelian Bruck loops (2007). Bruck loops can be studied with group theoretic methods, using the group $RMult(X) := < \rho_a: a \in X > \le Sym(X)$ with $\rho_a \in Sym(X), a \in X$ the right multiplications: $x \mapsto xa $. In joint work, B.Baumeister, A.Stein and G.Stroth analysed the structure of Bruck loops, extending results of M.Aschbacher, M.Kinyon and J.D.Phillips (2006).

---

Bipartite divisor graphs for group conjugacy class sizes

• 20 March, 2009, Blakers Lecture Theatre
• Daniela Bubboloni (Universita di Firenze)
  

We present various properties of a bipartite graph related to the sizes of the conjugacy classes of a finite group. Some invariants of the graph are rather strongly connected to the group structure. In particular the diameter is at most 6, and it is possible to classify those groups for which the graphs have diameter 6. If the graph is acyclic then the diameter is at most 5, and groups for which the graph is a path of length 5 are characterised.

---

The structure of the finite groups of conjugate rank two

• 6 March, 2009, Blakers Lecture Theatre
• Silvio Dolfi (Universita di Firenze)

In 1970 N. Ito proved that the finite groups with just three conjugacy class sizes (the groups of "conjugate rank two") are solvable groups. Improving on a result by A.R. Camina, in a joint work with E. Jabara, we determine their structure. They turn out to be either "F-groups" (i.e. groups in which the centralizers of noncentral elements are pairwise incomparable with respect to inclusion) or nilpotent groups. We can hence describe all possible configurations of conjugacy class sizes for groups of conjugate rank two.

---

Neighbour-transitive designs in odd graphs

• 27 February, 2009, Blakers Lecture Theatre
• Eugenia O'Reilley-Regueiro (Universidad Nacional Autónoma de México):
  

The odd graph O_k+1 is defined to be the graph whose vertices are the k-subsets of a set X of size 2k+1, and in which two such subsets are adjacent if and only if they are disjoint. It is a (k+1)-regular graph, and Aut(O_k+1)=S_2k+1. Now take U to be a subset of vertices in O_k+1, then G=Aut(U) is the stabiliser of U in S_2k+1. We consider the incidence structure (X,U) to be our design. We define U₁, the set of neighbours of U, as the set of vertices in O_k+1 that are NOT in U, and are adjacent to an element of U, and look for different choices of U such that G is neighbour transitive, that is, transitive on U₁. Cheryl E. Praeger and Robert A. Liebler have studied this problem with Johnson graphs instead of odd graphs, and Neil Gillespie is working on the analogous problem with Hamming graphs.

---

Antipodal coverings of classical distance regular graphs

• 17 February, 2009, Blakers Lecture Theatre
• Tony Gardiner (University of Birmingham)