Uniform Domination for Simple Groups
Algebra and Geometry Seminar
9th May 2018, 2:30 pm – 3:30 pm
Howard House, 4th Floor Seminar Room
It is well known that every finite simple group can be generated by just two elements. In fact, by a theorem of Guralnick and Kantor, there is a conjugacy class C such that for each non-identity element x there exists an element y in C such that x and y generate the entire group. Motivated by this, we introduce a new invariant for finite groups: the uniform domination number. This is the minimal size of a subset S of conjugate elements such that for each non-identity element x there exists an element s in S such that x and s generate the group. This invariant arises naturally in the study of generating graphs.
In this talk, I will present recent joint work with Tim Burness, which establishes best possible results on the uniform domination number for finite simple groups, using a mix of probabilistic and computational methods, together with recent results on the base sizes of primitive permutation groups.