EVENT: Groups and Combinatorics Afternoon

2pm Simon Guest (UWA/Baylor University)

Title: A Solvable Version of the Baer--Suzuki Theorem

Abstract: 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.

2:35 Akos Seress (Ohio State University/UWA)

Title: Majorana representations of dihedral, alternating, and symmetric groups



3:05 Afternoon Tea

3:40 Nicola Durante (Università di Napoli “Federico II,”)

Title: Buekenhout-Metz unitals

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

4:15 Frank De Clerck (Ghent University)

Title: A geometric approach to Mathon maximal arcs.

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.