Colva Roney-Dougal (St Andrews) : Generation of finite groups: from minimal generation to random subgroups

Friday 8 Dec 2017, 15:00 → 17:00 Europe/Brussels

CYCL 01 (Marc de Hemptinne)

It is a consequence of the classification of finite simple groups that every finite simple group can be generated by two elements. In a different direction, it was proved by Neumann that any subgroup of the symmetric group on n points can be generated by at most n/2 elements, except for the symmetric group S_3. This pair of results has been the starting point for a wealth of research on both the minimal number of generators for a finite group, and for the probability that k (uniform) random elements of a finite group G generate G. This talk will include a survey of some of this work, both classical and more recent. These results have interesting connections with the long-standing problem of determining properties of a uniform random subgroup of the symmetric group on n points.

pictures "800px-Symmetric_group_S4 Cayley_graph_of_Alternating_Group_A7.png