Generation of amalgamated free products of cyclic groups by finite automata over minimal alphabet

Abstract

Groups of automaton permutations over finite alphabets are considered. Finite automata over minimal possible alphabets that define amalgamated free products of finite cyclic groups are constructed. For any prime p the case of amalgamated free products of cyclic p-groups is considered. In this case it is shown that elements defined by constructed finite automata belong to p-Sylow subgroups of the group of automaton permutations over a p-element alphabet.