It is well-known that every finite simple group can be generated by two elements. Moreover, two arbitrary elements are very likely to generate the whole group. For example, every non-identity element of a finite simple group belongs to a generating pair. Groups with the latter property are said to be 3/2-generated. It is natural to ask which other finite groups are 3/2-generated. In 2008, Breuer, Guralnick and Kantor conjectured that a finite group is 3/2-generated if and only if every proper quotient of the group is cyclic. In this talk we will discuss recent progress towards establishing this conjecture, where probabilistic techniques play a key role. |