Texas A&M
University, Department of
Mathematics, 216 Milner Hall, 25th of February 2009, 3:00-3:50
Groups and Dynamics Seminar
The spread of a finite group
Simon Guest of Baylor University
Let G be a finite group. We say that G has spread at
least k if for any k distinct non-trivial elements
of G, x_1,...,x_k, there exists y in G such
that x_i and y generate G for every i=1,...,k. If G
does not have spread at least 1 then G is said to have spread 0.
We say that G has spread n if it has spread at least n
but it does not have spread at least n+1. Using elementary methods
we can prove that if G has a non-trivial normal subgroup N
such that G/N is non-cyclic then G must have spread 0. It has
been conjectured by Guralnick and Kantor that the converse is true.
They can prove that the converse holds in many cases. We will discuss
some recent joint work with Guralnick and Burness involving the
remaining cases.