What is the class of languages accepted by DFAs whose transition monoids are transitive permutation groups?
In the Wiki page A permutation automaton, or pure-group automaton, is a deterministic finite automaton such that each input symbol permutes the set of states. ..... A formal language is p-regular ...
Can the word problem in a free group be solved by a finite state automaton? I know it can be solved by a pushdown automaton.