A finitary automaton group is a generated by an invertible, deterministic finite-state letter-to-letter transducer whose only cycles are self-loops at identity state. We show that, for this presentation of finite groups, the uniform word problem coNP-complete. Here, input consists together with state sequence and question whether acts trivially on all words. Additionally, we also that respectiv...

in this paper‎, ‎we determine the simple groups $g=ab$‎, ‎where $b$ is isomorphic to $l_{3}(4)$ and $a$ isomorphic to an alternating or a symmetric group on $ngeq5$‎, ‎letters‎.