Authors : Md. Amzad Hossain; Md. Matiur Rahman; Md. Abdul Mannan

Volume/Issue : Volume 10 - 2025, Issue 4 - April

---

Google Scholar : https://tinyurl.com/3x7jtd4c

Scribd : https://tinyurl.com/ms5z6vrt

DOI : https://doi.org/10.38124/ijisrt/25apr199

Google Scholar

Note : A published paper may take 4-5 working days from the publication date to appear in PlumX Metrics, Semantic Scholar, and ResearchGate.

Note : Google Scholar may take 15 to 20 days to display the article.

---

Abstract : This paper aims at treating a study on the order of every element of 60 orders of group for multiplication composition. But the composition in G is associative; the multiplication composition is very significant in the order of elements of a group. We develop the order of a group, higher order of groups in different types of order and the order of elements of a group in real numbers. Let G be a group and let a n ∈ G be of infinite order n. In addition, notation o(a m) = λ m , where λ = l. c.m of m and n. If a ∈ G is of order n, then there exists an integer m for which a m = e if m is a multiple of n, in general we use this. Then we develop orders of elements of a cyclic group and every element of higher order of a group. After that we find out the order of every element of a group for the higher orders of the group for being binary operation.

Keywords : o(G), o(a), Multiplication Composition, LCM, Torsion Group.

References :

1. Marshall Hall Jr., David Wales, "The simple group of order 604, 800,” Journal of Algebra, Volume 9, Issue 4, Pages 417-450,August 1968. https://doi.org/10.1016/0021-8693 (68)90014-8
2. Brauer, R., Tuan, H.F, “On simple groups of finite order, “I Bulletin of the American Mathematical Society, Volume 51, Issue 10, Pages 756-766, October 1945. DOI: 10.1090/S0002-9904-1945-08441-9
3. Md. Abdul Mannan, Halima Akter and Samiran Mondal, “Evaluate All Possible Subgroups of a Group of Order 30 and 42 By Using Sylow’s Theorem, " International Journal of Scientific & Engineering Research, Volume 12, Issue 11, P 139-153, January-2021. ISSN 2229-5518,
4. Md. Abdul Mannan, Md. Amanat Ullah, Uttam Kumar Dey, Mohammad Alauddin , "A Study on Sylow Theorems for Finding out Possible Subgroups of a Group in Different Types of Order," Mathematics and Statistics, Vol. 10, No. 4, pp. 851 - 860, 2022. DOI: 10.13189/ms.2022.100416.
5. Mannan, M. A., Akter, H. ,& Ullah, . M. A. , " Evaluate All The Order of Every Element in The Higher Even, Odd, and Prime Order of Group for Composition, " Science and Technology Indonesia, 7(3), P 333–343, 2022. https://doi.org/10.26554/sti.2022.7.3.333-343
6. L. FINKELSTEIN AND A. RUDVALIS, “The Maximal Subgroups of Janko’s Simple Group of Order 50, 232, 960, "JOURNAL OF ALGEBRA, Volume 30, P122-143,1974.
7. J. MCKAY AND D. WALES, "The multipliers of the simple groups of order 604, 800 and 50, 232, 960, " J. Algebra, 17, P 262-273, 1971.
8. Ulderico Dardano, Silvana Rinauro, "Groups with many Subgroups which are Commensurable with some Normal Subgroup, " Advances in Group Theory and Applications, 7, PP 3–13, 2019. DOI: 10.32037/agta-2019-002
9. S. Trefethen, "Non-Abelian composition factors finite groups with the CUT-property,” J. Algebra, 522,P 236–242, 2019.
10. Mannan, Md.A., Nahar, N., Akter, H., Begum, M., Ullah, Md.A. and Mustari, S. "Evaluate All the Order of Every Element in the Higher Order of Group for Addition and Multiplication Composition, " International Journal of Modern Nonlinear Theory and Application, 11, P 11-30, 2022. https://doi.org/10.4236/ijmnta.2022.112002
11. D. J. S. Robinson, “A Course in the Theory of Groups,” Springer-Verlag, New York, 1982.
12. Brendan McCann, “On Products of Cyclic and Non-Abelian Finite p-Groups,” Advances in Group Theory and Applications, pp. 5–37,2020, ISSN: 2499-1287 DOI: 10.32037/agta-2020-001
13. D. Gorenstein, R. Lyons and R. Solomon, “The Classification of the Finite Simple Groups, Number 3,” Mathematical Surveys and Monographs, Volume 40, Amer. Math. Soc., 1998.
14. M. Hall, “On the Number of Sylow Subgroups in a Finite Group,” J. Algebra, 7, P 363–371,  1961.