Un théorème de Dixon pour les groupes d’automates

As every finite group is a subgroup of some permutation group, the most natural idea to generate random finite groups is to pick some random permutations and consider the group they generate. However, Dixon proved that this approach fails, as  it generically leads to generate the symmetric are the alternating group.

Mealy automata, on the other hand, are a powerful tool to generate groups and have been used to solve several important group theoretic conjectures. In particular, any finite group can be generated by a Mealy automaton, hence the idea of doing random generation by drawing random automata.

In this talk we will present this method and prove an analogue to Dixon’s theorem for a certain class of Mealy automata. Then we will discuss how to avoid this convergence of the distribution of generated groups.