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## On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups

#### Mourad CHELGHAM [1] , Mohamed KERADA [2] , Lemnouar NOUİ [3]

Let k>0 an integer. F, τ, N, Nk, and A denote, respectively, the classes of finite, torsion, nilpotent, nilpotent of class at most k, group in which every two generator subgroup is in Nk and abelian groups. The main results of this paper is, firstly, to prove that in the class of finitely generated FN-group, the property FC is closed under finite extension. Secondly, we prove that a finitely generated τN-group in the class ((τNk)τ,∞) ( respectively ((τNk)τ,∞)) is a τ-group (respectively τNc for certain integer c=c(k) ) and deduce that a finitely generated FN-group in the class ((FNk)F,∞) (respectively ((FNk)F,∞)) is -group (respectively FNc for certain integer c=c(k)). Thirdly we prove that a finitely generated NF-group in the class ((FNk)F,∞) ( respectively ((FNk)F,∞)) is F-group (respectively NcF for certain integer c=c(k)). Finally and particularly, we deduce that a finitely generated FN-group in the class ((FA)F,∞) (respectively ((FC)F,∞), ((FN)F,∞)) is in the class FA (respectively FN, FN(2)).

FC-group, (FC)F-group, (τNk)τ-group, (FNk)F-group, ((FNk)F, ∞)-group, ∞)∗-group, finitely generated group
• [1] A. Abdollahi et N. Trabelsi, Quelques extensions d'un problème de Paul Erdös sur les groupes, Bull. Belg. Math. Soc. 9, (2002), 1-11.
• [2] A. Abdollahi, Some Engel conditions on infinite subsets of certain groups, Bull.Austral. Math. Soc. 62, (2000), 141-148.
• [3] A. Abdollahi, Finitely generated soluble groups with an Engel conditions on infinite subsets, Rend. Sem. Mat. Univ. Padova. 103, (2000), 47-49.
• [4] A. Abdollahi, B. Taeri, A condition on finitely generated soluble groups, Communication in Algebra, 27(11), (1999), 5633-7658.
• [5] R. Baer, Finiteness properties of groups, Duke Math. J. 15, (1948), 1021-1032.
• [6] C. Casolo, Groups with all subgroups subnormal, Note Mat. 28, (2008), suppl. n. 2, 1-149.
• [7] C. Delizia and C. Nicotera, On residually finite groups with an Engel condition on infinitesubset, J. Austral. Math. Soc. (series A) 69, (2000), 415-420.
• [8] J. Erdös, The theory of groups with finite classes of conjugate elements, Acta. Math. Acad. Sci.Hungar. 5, (1954), 45-58.
• [9] F. Guerbi and T. Rouabeh, Hyper(Abelian-by-finite)-groups with many sebgroups of finite depth, 14(1), (2007), 17-28.
• [10] P. Hall, Finite-by-nilpotent-group, Prog. Cambridge Philos. Soc. 52, (1956), 611-616.
• [11] P. Hall, The Edmonton notes on nilpotent groups, Queen Mary CollegeMath. Notes (1969).
• [12] J. C. Lennox and J. Wiegold, Extensions of a problem of Paul Erdos on groups, J. Austral. Math. Soc. Ser. A, 31, (1981), 459-463.
• [13] B. H. Neumann, Groups with finite classes of conjugate elements, Proc. London. Math. Soc, (3.Ser.), 1, (1951), 178-187.
• [14] B. H. Neumann, A problem of Paul Erdös on groups, J. Austral. Math. Soc. ser. A 21, (1976), 467-472.
• [15] N. Nishigori, On some properties of FC-groups, J. Sci. Hiroshima Univ. Ser. A 21, (1957/1958), 99-105.
• [16] D. J. S. Robinson, Finiteness conditions and generalized soluble groups, Springer-Verlag, Berlin, Heidelberg, New York, (1972).
• [17] D. J. S. Robinson, A course in the theory of groups, Springer-verlag, Berlin, Heidelberg, New York, (1982.(
• [18] T. Rouabeh and N. Trabelsi, A note on Torsion-by-Nilpotent group, Rend. Sem. Mat. Univ. Panova, 117(2007), 175-179.
• [19] N. Trabelsi, Characterization of nilpotent-by-finite groups, Bull. Austral. Math. Soc, 61, (2000), 33-38.
• [20] N. Trabelsi, Finitely generated soluble groups with a condition on infinite subsets, Algebra Colloq, 9, (2002), 427-432.
• [21] N. Trabelsi, Soluble groups with many 2-generated torsion-by-nilpotent subgroups, Publ. Math. Debrecen, 67/1-2, 6, (2005), 93-102.
• [22] M. J. Tomkinson, FC-groups, Pitman Advanced Pub. Program, Californy university, USA, (1984.(
Primary Language en Research Article Author: Mourad CHELGHAM Author: Mohamed KERADA Author: Lemnouar NOUİ Publication Date : June 1, 2018
 Bibtex @research article { jnt431047, journal = {Journal of New Theory}, issn = {2149-1402}, eissn = {2149-1402}, address = {Mathematics Department, Gaziosmanpasa University 60250 Tokat-TURKEY.}, publisher = {Gaziosmanpasa University}, year = {2018}, volume = {}, pages = {22 - 30}, doi = {}, title = {On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups}, key = {cite}, author = {CHELGHAM, Mourad and KERADA, Mohamed and NOUİ, Lemnouar} } APA CHELGHAM, M , KERADA, M , NOUİ, L . (2018). On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups. Journal of New Theory , (23) , 22-30 . Retrieved from https://dergipark.org.tr/en/pub/jnt/issue/37237/431047 MLA CHELGHAM, M , KERADA, M , NOUİ, L . "On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups". Journal of New Theory (2018 ): 22-30 Chicago CHELGHAM, M , KERADA, M , NOUİ, L . "On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups". Journal of New Theory (2018 ): 22-30 RIS TY - JOUR T1 - On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups AU - Mourad CHELGHAM , Mohamed KERADA , Lemnouar NOUİ Y1 - 2018 PY - 2018 N1 - DO - T2 - Journal of New Theory JF - Journal JO - JOR SP - 22 EP - 30 VL - IS - 23 SN - 2149-1402-2149-1402 M3 - UR - Y2 - 2020 ER - EndNote %0 Journal of New Theory On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups %A Mourad CHELGHAM , Mohamed KERADA , Lemnouar NOUİ %T On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups %D 2018 %J Journal of New Theory %P 2149-1402-2149-1402 %V %N 23 %R %U ISNAD CHELGHAM, Mourad , KERADA, Mohamed , NOUİ, Lemnouar . "On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups". Journal of New Theory / 23 (June 2018): 22-30 . AMA CHELGHAM M , KERADA M , NOUİ L . On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups. JNT. 2018; (23): 22-30. Vancouver CHELGHAM M , KERADA M , NOUİ L . On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups. Journal of New Theory. 2018; (23): 30-22.

Authors of the Article
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