Russyev A. Finite subgroups and conjugacy in groups of finite automata

For the group of finite state automorphisms of a regular rooted tree it is determined conditions of finiteness of its self-similar subgroups and conditions of conjugacy for some of its elements. We have defined the class of automata without cycles with exit and proved that the group of an automaton without cycles with exit is finite. We have established the commutativity criterion of automaton group and proved that abelian group of an automaton is finite if and only if this automaton does not contain cycles with exit. For an automaton without cycles with exit with n states that satisfies special conditions we have proved that its action on n-th level of rooted tree is faithful. In case of binary alphabet we have found precise estimation of the order of automaton group as function of n. We have found complete list of pairwise non-isomorphic groups of automata without cycles with exit with 2, 3, 4 and 5 states. We have proved the conjugacy criterion of elements that have finite order and conjugacy criterion with adding machine over binary alphabet in the group of all finite-state automorphisms.