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Locally graded groups with a condition on infinite subsets | ||
| International Journal of Group Theory | ||
| مقاله 1، دوره 7، شماره 4، اسفند 2018، صفحه 1-7 اصل مقاله (203.76 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2016.21234 | ||
| نویسندگان | ||
| Asadollah Faramarzi Salles* ؛ Fatemeh Pazandeh Shanbehbazari | ||
| Damghan University | ||
| چکیده | ||
| Let $G$ be a group, we say that $G$ satisfies the property $\mathcal{T}(\infty)$ provided that, every infinite set of elements of $G$ contains elements $x\neq y, z$ such that $[x, y, z]=1=[y, z, x]=[z, x, y]$. We denote by $\mathcal{C}$ the class of all polycyclic groups, $\mathcal{S}$ the class of all soluble groups, $\mathcal{R}$ the class of all residually finite groups, $\mathcal{L}$ the class of all locally graded groups, $\mathcal{N}_2$ the class of all nilpotent group of class at most two, and $\mathcal{F}$ the class of all finite groups. In this paper, first we shall prove that if $G$ is a finitely generated locally graded group, then $G$ satisfies $\mathcal{T}(\infty)$ if and only if $G/Z_2(G)$ is finite, and then we shall conclude that if $G$ is a finitely generated group in $\mathcal{T}(\infty)$, then \[G\in\mathcal{L}\Leftrightarrow G\in\mathcal{R}\Leftrightarrow G\in\mathcal{S}\Leftrightarrow G\in\mathcal{C}\Leftrightarrow G\in\mathcal{N}_2\mathcal{F}.\] | ||
| کلیدواژه‌ها | ||
| Finitely generated groups؛ Residually finite groups؛ Locally graded groups | ||
| مراجع | ||
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[1] A. Abdollahi, Finitely generated soluble groups with an Engel condition on infinite subsets, Rend. Sem. Mat. Univ. Padova, 103 (2000) 47–49.
[2] A. Abdollahi and B. Taeri, A condition on finitely generated soluble groups, Comm. Algebra, 27 (1999) 5633–5638.
[3] C. Delizia, Finitely generated soluble groups with a condition on infinite subsets, Istit. Lombardo Accad. Sci. Lett. Rend. A, 128 (1994) 201–208.
[4] C. Delizia, On certain residually finite groups, Comm. Algebra, 24 (1996) 3531-3535.
[5] C. Delizia and C. Nicotera, Groups with conditions on infinite subsets, Ischia Group Theory 2006: Proceedings of a Conference in Honor of Akbar Rhmetulla, World Scientific Publishing, Singapore, 2007 46–55.
[6] C. Delizia, A. Rhemtulla and H. Smith, Locally graded groups with a nilpotency condition on infinite subsets, J. Austral. Math. Soc. Ser. A, 69 (2000) 415–420.
[7] J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic pro-p-groups, London Math. Soc. Lecture Note Series, 157, Cambridge Univ. Press, Cambridge, 1991.
[8] A. Faramarzi Salles, Finitely generated soluble groups with a condition on infinite subsets, Bull. Aust. Math. Soc., 87 (2013) 152–157.
[9] Y. K. Kim and A. H. Rhemtulla, Weak maximality condition and polycyclic groups, Proc. Amer. Math. Soc., 123 (1995) 711–714.
[10] J. C. Lennox and J. Wiegold, Extensions of a problem of Paul Erdös on groups, J. Austral. Math. Soc. Ser. A, 31 (1981) 459–463.
[11] P. Longobardi, On locally graded groups with an Engel condition on infinite subsets, Arch. Math. (Basel), 76 (2001) 88–90.
[12] B. H. Neumann, A problem of Paul Erdös on groups, J. Austral. Math. Soc. Ser. A, 21 (1976) 467–472.
[13] D. J. Robinson, A course in the theory of groups, Second Edition, Springer-Verlag, Berlin, 1982.
[14] J. Tits, Free subgroups in linear groups, J. Algebra, 20 (1972) 250–270. | ||
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