On Some Applications and Open Problems about (m-Groups)

Oliver V. Shtawzen

doi:https://doi.org/10.54216/GJMSA.010105

On Some Applications and Open Problems about (m-Groups)

Oliver V. Shtawzen ^{1
*}

1 University Of Nizwa, Sultanate Of Oman - (Vonshtawzen1970abc@gmail.com)

Doi: https://doi.org/10.54216/GJMSA.010105

Received: February 20, 2022 Accepted: May 12, 2022

Abstract

The generalizations of abelian groups have been studied widely because of their importance in classification theorem and representation. A group G is called an m-power closed group or (m-group) if and only if it has the following property x^m y^m=z^m ∀x,y ∈ G and for z ∈ G. This paper studies a special case of m-groups, when G is a finite m-group and n-group at the same time with relatively prime integers m and n, which is called a Monic group. It presents the necessary and sufficient conditions for a monic group G to be cyclic, abelian, nilpotent, and solvable by the corresponding property of its power subgroups G_m, G_n. Also, this work introduces three open problems in the theory of finite groups.

Keywords :

m-power closed group , m-abelian group , monic group.

References

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[6] Arad, Z. and Ward, M. New Criteria for the Solvability of Finite Groups, Journal of Algebra, 77, (1982), pp. 234-246.

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[8] Sankari, H., and Abobala, M., On A New Criteria For the Solvability of Non-Simple Finite Groups and m-Abelian Solvability, Journal of Mathematics, Hindawi, 2021.

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Cite This Article As :

V., Oliver. On Some Applications and Open Problems about (m-Groups). Galoitica: Journal of Mathematical Structures and Applications, vol. , no. , 2022, pp. 48-51. DOI: https://doi.org/10.54216/GJMSA.010105

V., O. (2022). On Some Applications and Open Problems about (m-Groups). Galoitica: Journal of Mathematical Structures and Applications, (), 48-51. DOI: https://doi.org/10.54216/GJMSA.010105

V., Oliver. On Some Applications and Open Problems about (m-Groups). Galoitica: Journal of Mathematical Structures and Applications , no. (2022): 48-51. DOI: https://doi.org/10.54216/GJMSA.010105

V., O. (2022) . On Some Applications and Open Problems about (m-Groups). Galoitica: Journal of Mathematical Structures and Applications , () , 48-51 . DOI: https://doi.org/10.54216/GJMSA.010105

V. O. [2022]. On Some Applications and Open Problems about (m-Groups). Galoitica: Journal of Mathematical Structures and Applications. (): 48-51. DOI: https://doi.org/10.54216/GJMSA.010105

V., O. "On Some Applications and Open Problems about (m-Groups)," Galoitica: Journal of Mathematical Structures and Applications, vol. , no. , pp. 48-51, 2022. DOI: https://doi.org/10.54216/GJMSA.010105

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On Some Applications and Open Problems about (m-Groups)

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