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Dixon's famous theorem states that the group generated by two randompermutations of a finite set is generically either the whole symmetric group orthe alternating group. In the context of random generation of finite groupsthis means that it is hopeless to wish for a uniform distribution -- or even anon-trivial one -- by drawing random permutations and looking at the generatedgroup. Mealy automata are a powerful tool to generate groups, including allfinite groups and many interesting infinite ones, whence the idea of generatingrandom finite groups by drawing random Mealy automata. In this paper we showthat, for a special class of Mealy automata that generate only finite groups,the distribution is far from being uniform since the obtained groups aregenerically a semi-direct product between a direct product of alternatinggroups and a group generated by a tuple of transpositions.

An analogue to Dixon's theorem for automaton groups