Abstract
Systems of induced generating actions of automaton permutations groups on words of length r are investigated. A family of irreducible systems of generators is constructed, and the cardinality of such systems is found to be related with r. The infinite generativity (even in the topological sense) of groups of all automaton permutations, finite automaton permutations, and finitary automaton permutations is established; the well-known proof of this fact contained an error.
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References
V. M. Glushkov, “Abstract theory of automata,” Usp. Mat. Nauk,16, No. 5 (101), 3–62 (1961).
Ya. Khoreys, “Transformations determined by finite automata,” Probl. Kibern., No. 9, 23–26 (1963).
V. P. Zarovnyi, “A group of automaton one-to-one mappings,” Dokl. Akad. Nauk SSSR,156, No. 6, 1266–1269 (1964).
V. P. Zarovnyi, “Automaton permutations and wreath products of groups,” Dokl. Akad. Nauk SSSR,160, No. 3, 562–565 (1965).
V. P. Zarovnyi, “Automaton permutations and wreath products of groups,” Kibernetika, No. 1, 29–36 (1965).
F. Gecseg, “A group of one-to-one transformations determined by finite automata,” Kibernetika, No. 1, 37–39 (1965).
B. Chakan’ and F. Gecseg, “A group of automaton permutations,” Kibernetika, No. 5, 14–17 (1965).
F. Gecseg and I. Peak, Algebraic Theory of Automata, Akademiai Kiado, Budapest (1972).
S. V. Alyoshin, “Finite automata and the Burnside problem on periodic groups,” Mat. Zametki,11, No. 3, 319–328 (1972).
V. I. Sushchanskii, “Periodic permutation P-groups and the Burnside unrestricted problem,” Dokl. Akad. Nauk SSSR,20, No. 4, 766–770 (1979).
R. I. Grigorchuk, “On the Burnside problem on periodic groups,” Funkts. Anal. Prilozh.,14, No. 1, 53–54 (1980).
N. Gupta and S. Sidki, “On the Burnside problem for periodic groups,” Math. Z.,182, 385–388 (1982).
V. I. Sushchanskii, “Normal structure of groups of isometries of semifinite Baer metrics,” in: Infinite Groups and Related Algebraic Structures, Inst. Mat. NANU, Kiev (1993), pp. 269–289.
Said Sidki, Regular Trees and Their Automorphism, Monografias de Matematica, Vol. 56, IMPA, Rio de Janeiro (1998).
V. Sushchanskii, “Groups of automaton permutations,” Dop. NAN Ukr., No. 6, 47–51 (1998).
V. Sushchanskii, “Groups of finite automaton permutations,” Dop. NAN Ukr., No. 2, 29–32 (1999).
I. D. Ivanyuta, “SylovP-subgroups of a countable symmetric group,” Ukr. Mat. Zh.,15, No. 3, 240–249 (1963).
L. A. Kaluzhnin, “A generalization of SylovP-subgroups of symmetric groups,” Acta Math. Hung.,2, Nos. 3–4, 198–221 (1951).
A. G. Kurosh, Group Theory [in Russian], Nauka, Moscow (1972).
G. Raney, “Sequential functions,” Kibern. Sb., No. 3, 45–49 (1961).
V. S. Sikora, “Diameter of the Cayley graph of the wreath product of two symmetric groups for a two-element system of generators,” Nauk. Visnyk Chernivets. Univ., Ser. Matematika, No. 76, 99–105, Ruta, Chernovtsy (2000).
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Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 121–133, May–June, 2000.
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Sikora, V.S., Sushchanskii, V.I. Systems of generators of automaton permutations groups. Cybern Syst Anal 36, 415–425 (2000). https://doi.org/10.1007/BF02732992
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DOI: https://doi.org/10.1007/BF02732992