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International Journal of Neutrosophic Science
Volume 19 , Issue 1, PP: 384-388 , 2022 | Cite this article as | XML | Html |PDF

Title

The NILPOTENT Characterization of the finite neutrosophic p-groups

  S. A. Adebisi 1 * ,   Florentin Smarandache 2

1  Department of Mathematics, Faculty of Science, University of Lagos, Nigeria
    (adesinasunday@yahoo.com)

2  University of New Mexico, Gallup Campus, NM 87301, USA
    (smarand@unm.edu)


Doi   :   https://doi.org/10.54216/IJNS.190134

Received: March 21, 2022 Accepted: September 05, 2022

Abstract :

A well known and referenced global result is the nilpotent characterisation of the finite p-groups. This undoubtedly transends into neutrosophy. Hence, this fact of the neutrosophic nilpotent p-groups is worth critical studying and comprehensive analysis. The nilpotent characterisation depicts that there exists a derived series (Lower Central) which must terminate at {ϵ} ( an identity ) , after a finite number of steps. Now, Suppose that G(I) is a neutrosophic p-group of class at least m ≥ 3. We show in this paper that Lm−1(G(I)) is abelian and hence G(I) possesses a characteristic abelian neutrosophic subgroup which is not supposed to be contained in Z(G(I)). Furthermore, If L3(G(I)) = 1 such that pm is the highest order of an element of G(I)/L2(G(I)) (where G(I) is any neutrosophic p-group) then no element of L2(G(I)) has an order higher than pm.

Keywords :

Neutrosophic p-groups ; Nilpotency; central series , order; commutator; abelian

References :

[1] A.A.A. Agboola, On Refined Neutrosophic Quotient Groups, International Journal of Neutrosophic

Science (IJNS) Vol. 5, No. 2, pp. 76-82, 2020 (Doi :10.5281/zenodo.3828609)

[2] A.A.A. Agboola, On Refined Neutrosophic Algebraic Structures, Neutrosophic Sets and Systems,

vol.10, pp. 99-101, 2015.

[3] E.O. Adeleke, A.A.A. Agboola and F. Smarandache, Refined Neutrosophic Rings I, International

Journal of Neutrosophic Science (IJNS), vol. 2(2), pp. 77-81, 2020. (DOI:10.5281/zenodo.3728222)

[4] Florentin Smarandache, Madeline Al-Tahan , Theory and Applications of NeutroAlgebras as Generalizations

of Classical Algebras IGI Global

[5] H. Marshall, (1959). The Theory of Groups. The Macmillan Company, New York.

[6] G. Frobenius, and L. Stickelberger, (1879). ¨Uber Gruppen von Vertauschbaren Elementen, J.

reine angew. Math. 86 (217-262).

[7] A. O. Kuku, (1992). Abstract Algebra, Ibadan University Press.

[8] S. Mattarei, (1994). An example of p-groups with identical character tables and different derived

length. Arch. Math 62 (12-20).

[9] H. Wielandt, (1959). Ein Beweis f¨ur die Existenz der sylowgruppen. Arch. Math. 10 (401-402).

(MR 26#504).

[10] A. Weir (1955). Sylow p-subgroups of the classical groups over finite fields with characteristic

prime to p. Proc. Amer. Math. Soc. 6, (529-533).

[11] A. Weir, (1955). The sylow subgroups of the symmetric groups. Proc. Amer. Math. Soc. 6

(534-541).


Cite this Article as :
Style #
MLA S. A. Adebisi, Florentin Smarandache. "The NILPOTENT Characterization of the finite neutrosophic p-groups." International Journal of Neutrosophic Science, Vol. 19, No. 1, 2022 ,PP. 384-388 (Doi   :  https://doi.org/10.54216/IJNS.190134)
APA S. A. Adebisi, Florentin Smarandache. (2022). The NILPOTENT Characterization of the finite neutrosophic p-groups. Journal of International Journal of Neutrosophic Science, 19 ( 1 ), 384-388 (Doi   :  https://doi.org/10.54216/IJNS.190134)
Chicago S. A. Adebisi, Florentin Smarandache. "The NILPOTENT Characterization of the finite neutrosophic p-groups." Journal of International Journal of Neutrosophic Science, 19 no. 1 (2022): 384-388 (Doi   :  https://doi.org/10.54216/IJNS.190134)
Harvard S. A. Adebisi, Florentin Smarandache. (2022). The NILPOTENT Characterization of the finite neutrosophic p-groups. Journal of International Journal of Neutrosophic Science, 19 ( 1 ), 384-388 (Doi   :  https://doi.org/10.54216/IJNS.190134)
Vancouver S. A. Adebisi, Florentin Smarandache. The NILPOTENT Characterization of the finite neutrosophic p-groups. Journal of International Journal of Neutrosophic Science, (2022); 19 ( 1 ): 384-388 (Doi   :  https://doi.org/10.54216/IJNS.190134)
IEEE S. A. Adebisi, Florentin Smarandache, The NILPOTENT Characterization of the finite neutrosophic p-groups, Journal of International Journal of Neutrosophic Science, Vol. 19 , No. 1 , (2022) : 384-388 (Doi   :  https://doi.org/10.54216/IJNS.190134)