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.

中文翻译：

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最小字母上的有限自动机生成循环群的合并自由产物

考虑了有限字母上的自动机排列组。构造了最小可能字母上的有限自动机，这些字母定义了有限循环组的合并自由乘积。对于任何素数p，都考虑了环状p-基团的自由结合产物的情况。在这种情况下，表明由构造的有限自动机定义的元素属于p元素字母上的自动机排列组的p -Sylow子组。