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Centre for the Mathematics of Symmetry and Computation
News and events

Centre for the Mathematics of Symmetry and Computation

News and events

Seminars

Previous seminars co-ordinated by the Groups and Combinatorics research group during 2010:

Hemisystems of generalised quadrangles

12noon, 2 March 2010, Maths Lecture Room 2
John Bamberg (University of Western Australia)

(Joint work with Michael Giudici and Gordon Royle). A hemisystem of a generalised quadrangle is a set of half the lines of the generalised quadrangle such that every point is on a constant number m of lines. (So necessarily, m is half the number of lines on any point). The generalised quadrangles of interest are those which meet the Higman-Sims bound, whereby a hemisystem naturally produces a partial quadrangle and strongly regular graph. Segre (1965) showed that there exists a hemisystem of the classical generalised quadrangle H(3,3²), and it was conjectured by J.A. Thas in 1995 that no hemisystem of H(3,q²) exists for q>3. Ten years later, Cossidente and Penttila proved that for every q odd, there exists a hemisystem of H(3,q²), and their examples arise from an embedding of a sub-geometry into H(3,q²) (namely, an elliptic quadric Q-(3,q)). The only known generalised quadrangles with the same parameters of H(3,q²), q odd, are the flock generalised quadrangles. We will present in this talk an improvement of Cossidente and Penttila's results to flock generalised quadrangles.

On a problem of Janko, a conjecture of Isbell and a 50 pound prize of Cameron

12noon, 23 February 2010, Maths Lecture Room 2
Pablo Spiga (University of Western Australia)

Recently Janko asked whether it is possible to classify the 2-generated p-groups whose maximal subgroups are 2-generated. In this talk we present some data and some partial results in order to show that a possible classification seems feasible for p>4. Also, we show how the groups arising in this classification are useful for obtaining a partial result towards a conjecture of Isbell (and hopefully a 50 pound prize of Cameron). All required definitions are given during the talk.

Some results on constant maps in the transition monoid of an automaton

12noon, 16 February 2010, Maths Lecture Room 2
Tim Boykett (Johannes Kepler Universitat Linz, Austria)

We investigate finite state automata and are interested in the situation when certain words can map any given state to a fixed terminal state. These so called reset words correspond to constant maps or right zeroes in the transition monoid of the automaton. A conjecture of J. Cerny states that if an automaton has a reset word, then it has one of length less than or equal to (n-1)^2, where n is the number of states. This bound is reached by a class defined by Cerny and 8 known sporadic examples. In this talk I will present an outline of some results including some classes with much smaller minimal reset words. I will also outline several open areas of work.

Construction of Large Graphs: Few Results and a Wealth of Open Problems

11am, 25 January 2010, Maths Lecture Room 2
Guillermo Pineda-Villavicencio (University of Ballarat)

In recent work with Eyal Loz we constructed all the new largest known graphs of maximum degree 16< k < 21 and diameter 1< D < 11, using a wide range of techniques and concepts, such as graph compounding, vertex duplication, Kronecker product, polarity graphs and voltage graphs. In this way, we extended the table of largest graphs known at present, which previously only covered graphs of maximum degree 2< k< 17 and diameter 1< D < 11.

During the constructing process we arrived at the conclusion that, among all the techniques employed, the notion of voltage graphs is the leading beacon for future directions in the construction of large graphs. However, many of its weaknesses need to be tackle to reach its potential. For instance, (i) There is NO methodology to obtain voltage groups and voltage assignments to generate graphs with specific properties. (ii) The procedure is done "blindly" in some sense; little is known about the classes of graphs that can be lifted.

In our talk we present a number of ideas and open problems that address the previous defects; some of the open problems are related below. Problem 1: Develop a methodology to construct the voltage group and the voltage assignment, given some desired properties for the lift. Problem 2: Given a base graph with a graph property P, provide conditions for a lift to have the property P. (Special case of the previous problem).

Problem 3: For what graphs (not necessarily vertex-transitive) G, does Aut(G) contain a subgroup semiregular on V(G)?. This is joint work with Eyal Loz and Hebert Peerez-Roses

A Solvable Version of the Baer--Suzuki Theorem

2pm, January 14 2010
Simon Guest (UWA/Baylor University)

Let G be a finite group, and take an element x in G. The Baer--Suzuki states that if every pair of conjugates of x generates a nilpotent group then the group generated by all of the conjugates of x is nilpotent. It is natural to ask if an analogous theorem is true for solvable groups. Namely, if every pair of conjugates of x generates a solvable group then is the group generate by all of the conjugates of x solvable? In fact, this is not true. For example, if x has order 2 in a (nonabelian) simple group G then every pair of conjugates of x generates a dihedral group (which is solvable), but the normal subgroup generated by all of the conjugates of x must be the whole of the nonabelian simple group G, which of course is not solvable.

There are also counterexamples when x has order 3. However, the following is true: (1) Let x in G have prime order p > 4. If every pair of conjugates of x generates a solvable group then the group generated by all of the conjugates of x is solvable. (2) Let x in G be an element of any order. If every 4-tuple of conjugates x, x^{g_1}, x^{g_2}, x^{g_3} generates a solvable group then the group generated by all of the conjugates of x is solvable.. We will discuss these results, some generalizations, and some of the methods used in their proof.

Majorana representations of dihedral, alternating, and symmetric groups

2.35pm, January 14 2010
Akos Seress (Ohio State University/UWA)

Buekenhout-Metz unitals

3.40pm, January 14 2010
Nicola Durante (Università di Napoli "Federico II")

We will discuss on some recent characterization theorems for Buekenhout-Metz unitals in a Desarguesian projective plane of square order.

A geometric approach to Mathon maximal arcs

4.15pm, January 14 2010
Frank De Clerck (Ghent University)

A maximal arc of degree d in a projective plane of order q is a non-empty, proper subset of points such that every line meets the set in 0 or d points, for some d. If a plane has a maximal arc of degree d the dual plane has one of degree q/d. We will mainly restrict to Desarguesian planes. It has been proved by Ball, Blokhuis and Mazzocca that non-trivial maximal arcs in PG(2,q) can not exist if q is odd. They do exist if q is even: examples are hyperovals, Denniston arcs, Thas arcs and Mathon arcs. We will give an overview of these constructions and of the connection with other geometric topics. We will give a geometric approach to the Mathon arcs emphasising on those of degree 8.

The University of Western Australia