Generators and relations for finite simple groupsAlgebra & Discrete Mathematics
|Robert Guralnick, University of Southern California
|Fri, Mar 21 2008, 2:10PM
It is known that every finite simple group can be generated by two elements. More recently, there have been probabilistic approaches to this -- most pairs of elements generate the simple group. We will discuss some variations on this -- for example, fix a nontrivial element in the simple group; can you generate the group with one more element? The other question we will discuss is how many relations are required to present the simple group. The outstanding conjecture had been that one needed log |G| relations. Recent work of Guralnick, Kantor, Kassabov and Lubotzky shows that 80 relations suffice except for possibly one family of simple groups. One can also ask about the lengths of the relations involved in the presentation.