Abstract
Some problems that in general have a negative answer have an affirmative answer in the class of locally graded groups and a negative answer outside of this class. We present three such problems and mention three others, which possibly are of that type.
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References
S. I. Adian, The Problem of Burnside and Identities in Groups [in Russian], Nauka, Moscow (1975). See also trans. J. Lennox and J. Wiegold, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92, Springer, Berlin (1979).
B. Bajorska and O. Macedońska, “On positive law problems in the class of locally graded groups,” Comm. Algebra, 32, No. 5, 1841–1846 (2004).
G. Bergman, “Hyperidentities of groups and semigroups,” Aequationes Math., 23, 55–65 (1981).
G. Bergman, Questions in Algebra, preprint, Berkeley, 1986.
M. Boffa, “Elimination of inverses in groups,” in: Model Theory of Groups and Automorphism Groups, London Math. Soc. Lecture Notes Series, Vol. 224, 1997, pp. 134–143.
R. G. Burns and Yu. Medvedev, “A note on Engel groups and local nilpotence,” J. Austral. Math. Soc., 64, 92–100 (1998).
R. G. Burns and Yu. Medvedev, “Group laws implying virtual nilpotence,” J. Austral. Math. Soc., 74, 295–312 (2003).
S. N. Chernikov, “Infinite non-Abelian groups with an invariance condition for infinite non-Abelian subgroups,” Dokl. Akad. Nauk SSSR, 194, 1280–1283 (1970).
M. Gromov, “Groups of polynomial growth and expanding maps,” in: Publs. Math. Inst. Hautes Étud. Sci., Vol. 53, 1981, pp. 53–73.
J. R. J. Groves, “Varieties of soluble groups and a dichotomy of P. Hall,” Bull. Austral. Math. Soc., 5, 391–410 (1971).
Y. Kim and A. H. Rhemtulla, “On locally graded groups,” in: Groups-Korea’94, Walter de Gruyter, Berlin (1995), pp. 189–197.
The Kourovka Notebook: Unsolved Problems in Group Theory, 12th ed., Inst. Mat. Sib. Otdel. Akad. Nauk Rossii, Novosibirsk (1993).
P. Kozhevnikov and O. Macedońska, “On varieties of groups without positive laws,” Comm. Algebra, 30, No. 9, 4331–4334 (2002).
J. Lewin and T. Lewin, “Semigroup laws in varieties of soluble groups,” Math. Proc. Cambridge Philos. Soc., 65, 1–9 (1969).
O. Macedońska, “Groupland,” in: C. Campbell, E. Robertson, and G. Smith, eds., Groups St Andrews 2001 in Oxford, Cambridge Univ. Press, Cambridge (2003), pp. 400–404.
A. I. Mal’cev, “Nilpotent semigroups,” Uchen. Zap. Ivanovsk. Ped. Inst., 4, 107–111 (1953).
J. Milnor, “Growth of finitely generated solvable groups,” J. Differential Geom., 2, 447–449 (1968).
B. H. Neumann and T. Taylor, “Subsemigroups of nilpotent groups,” Proc. Roy. Soc. London Ser. A, 274, 1–4 (1963).
S. Oates and M. B. Powell, “Identical relations in finite groups,” J. Algebra, 1, 11–39 (1964).
A. Yu. Ol’shanskii, Geometry of Defining Relations in Groups, Kluwer Academic, Dordrecht (1991).
A. Yu. Ol’shanskii and A. Storozhev, “A group variety defined by a semigroup law,” J. Austral. Math. Soc. Ser. A, 60, 255–259 (1996).
A. Shalev, “Combinatorial conditions in residually finite groups. II,” J. Algebra, 157, 51–62 (1993).
L. N. Shevrin and M. V. Volkov, “Semigroup identities,” Izv. Vyssh. Uchebn. Zaved. Mat., 11, 3–47 (1985).
G. Traustason, “Semigroup identities in 4-Engel groups,” J. Group Theory, 2, 39–46 (1999).
M. Vaughan-Lee, “On Zelmanov’s solution of the restricted Burnside problem,” J. Group Theory, 1, 65–94 (1998).
J. A. Wolf, “Growth of finitely generated solvable groups and curvature of Riemannian manifolds,” J. Differential Geom., 2, 421–446 (1968).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 127–133, 2005.
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Macedońska, O. On difficult problems and locally graded groups. J Math Sci 142, 1949–1953 (2007). https://doi.org/10.1007/s10958-007-0102-9
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DOI: https://doi.org/10.1007/s10958-007-0102-9