Abstract:
Studying generating sets for groups has led to many interesting and surprising results. For instance, every finite simple group can be generated by just two elements. In fact, Guralnick and Kantor, in 2000, proved that in a finite simple group every nontrivial element is contained in a generating pair, a property known as 3/2-generation. This answers a 1962 question of Steinberg. In this talk I will report on recent progess towards a classification of the finite 3/2-generated groups, and I will discuss joint work with Casey Donoven where we found the first nontrivial examples of infinite 3/2-generated groups.