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On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FN<sub>k</sub>-groups

Year 2018, Issue: 23, 22 - 30, 01.06.2018

Abstract

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)).

References

  • [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.(
Year 2018, Issue: 23, 22 - 30, 01.06.2018

Abstract

References

  • [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.(
There are 22 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Mourad Chelgham This is me

Mohamed Kerada This is me

Lemnouar Noui This is me

Publication Date June 1, 2018
Submission Date March 11, 2018
Published in Issue Year 2018 Issue: 23

Cite

APA Chelgham, M., Kerada, M., & Noui, L. (2018). On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups. Journal of New Theory(23), 22-30.
AMA Chelgham M, Kerada M, Noui L. On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups. JNT. June 2018;(23):22-30.
Chicago Chelgham, Mourad, Mohamed Kerada, and Lemnouar Noui. “On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-Groups”. Journal of New Theory, no. 23 (June 2018): 22-30.
EndNote Chelgham M, Kerada M, Noui L (June 1, 2018) On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups. Journal of New Theory 23 22–30.
IEEE M. Chelgham, M. Kerada, and L. Noui, “On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups”, JNT, no. 23, pp. 22–30, June 2018.
ISNAD Chelgham, Mourad et al. “On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-Groups”. Journal of New Theory 23 (June 2018), 22-30.
JAMA Chelgham M, Kerada M, Noui L. On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups. JNT. 2018;:22–30.
MLA Chelgham, Mourad et al. “On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-Groups”. Journal of New Theory, no. 23, 2018, pp. 22-30.
Vancouver Chelgham M, Kerada M, Noui L. On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups. JNT. 2018(23):22-30.


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