EVENT: Groups and Combinatorics Afternoon


2pm Simon Guest (UWA/Baylor University)
Title: A Solvable Version of the BaerSuzuki Theorem
Abstract: Let G be a finite group, and take an element x in G. The BaerSuzuki 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 4tuple 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: BuekenhoutMetz unitals
Abstract: We will discuss on some recent characterization theorems for BuekenhoutMetz 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 nonempty, 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 nontrivial 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.
Location 
Weatherburn Lecture Theatre


Contact 
Michael Giudici
<[email protected]>

Start 
Thu, 14 Jan 2010 14:00

End 
Thu, 14 Jan 2010 17:00

Submitted by 
Michael Giudici <[email protected]>

Last Updated 
Tue, 12 Jan 2010 11:09

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