It is well known that many famous Burnside-type problems have positive solutions for PI-groups and PI-algebras. In the present article, we also consider various Burnside-type problems for PI-groups and PI-representations of groups.
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References
E. S. Golod and I. R. Shafarevich, “On the class field tower,” Izv. Akad. Nauk SSSR, Ser. Mat., 28, 261–272 (1964).
K. A. Hirsch, “Über lokal-nilpotente Gruppen,” Math. Z., 63, 290–294 (1955).
N. Jacobson, Structure of Rings, Amer. Math. Soc., Providence (1964).
L. Kaloujnine, “Über gewisse Beziehungen zwischen einer Gruppe und ihren Automorphismen,” Berliner Math. Tagung, 164–172 (1953).
E. R. Kolchin, “On certain concepts in the theory of algebraic matrix groups,” Ann. Math., 49, 774–789 (1948).
P. S. Novikov and S. I. Adian, “On infinite periodic groups. I, II, III,” Izv. Akad. Nauk SSSR, Ser. Mat. 32, 212–244, 251–524, 709–731 (1968).
S. A. Pikhtilkov, “On the prime radical of PI-representable groups,” Math. Notes, 72, No. 5–6, 682–686 (2002).
V. P. Platonov, “Engel elements and the radical in PI-algebras and topological groups,” Sov. Math. Dokl., 6, 412–415 (1965).
B. I. Plotkin, “On some criteria of locally nilpotent groups,” Am. Math. Soc. Transl., 17, 1–7 (1961).
B. I. Plotkin, Group of Automorphisms of Algebraic System, Wolters-Noordhoff, Groningen (1972).
B. I. Plotkin, “Notes on Engel groups and Engel elements in groups. Some generalizations,” Izv. Ural. Gos. Univ. Mat. Mekh., 36, No. 7, 153–166, 192–193 (2005).
B. I. Plotkin and S. M. Vovsi, Varieties of Group Representations. General Theory, Connections, and Applications [in Russian], Zinatne, Riga (1983).
C. Procesi, “The Burnside problem,” J. Algebra, 4, 421–425 (1966).
J. Tits, “Free subgroups in linear groups,” J. Algebra, 20, 250–270 (1972).
A. Tokarenko, “On linear groups over rings,” Sib. Mat. Zh., 9, No. 4, 951–959 (1968).
O. Zariski and P. Samuel, Commutative Algebra, Vol. 1, Univ. Ser. Higher Math., Van Nostrand, Princeton (1958).
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Dedicated to the memory of the outstanding person and teacher Alexander Gennadievich Kurosh
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 14, No. 7, pp. 3–13, 2008.
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Aladova, E., Plotkin, B. PI-groups and PI-representations of groups. J Math Sci 164, 155–162 (2010). https://doi.org/10.1007/s10958-009-9745-z
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DOI: https://doi.org/10.1007/s10958-009-9745-z