# Evaluate All The Order of Every Element in The Higher Even, Odd, and Prime Order of Group for Composition

## Abstract

This paper aims to treat a study on the order of every element in the higher even, odd and prime order of group for composition. In fact, express order of a group and order of an element of a group in real numbers. Here we discuss the higher order of groups in different types of order, which will give us practical knowledge to see the applications of the composition. In order to find out the order of an element amG in which an= e= identity element, then find the least common multiple (i.e.(LCM))= λ) of m and n. The least common multiple of two numbers is the "smallest non-zero common number," which is a multiple of both the numbers. So O(am )= λ/m. Also, if G is a finite group, n is a positive integer, and aG then the order of the products na. When G is a finite group, every element must have finite order, but the converse is false. There are infinite groups where each element has finite order. Finally, find out the order of every element of a group in different types of the higher even, odd and prime order of group for composition.

## References

Bächle, A. (2020). 3 Questions on Cut Groups. arXiv preprint arXiv:2001.02637

Beltrán, A., R. Lyons, A. Moretó, G. Navarro, A. Sáez, and P. H. Tiep (2019). Order of Products of Elements in Finite Groups. Journal of The London Mathematical Society, 99(2); 535–552

Brauer, R. (1942). On Groups whose Order Contains a PrimeNumber to The First Power I. American Journal of Mathematics, 64(1); 401–420

Cavaleri, M., D. D’Angeli, A. Donno, and E. Rodaro (2021). Graph Automaton Groups. Advances in Group Theory and Applications, 11; 75–112

Cook, W. J., J. Hall, V. W. Klima, and C. Murray (2019). Leibniz Algebras with Low-Dimensional Maximal Lie Quotients. Involve, a Journal of Mathematics, 12(5); 839–853

d’Angeli, D., D. Francoeur, E. Rodaro, and J. P. Wächter (2020). Infinite Automaton Semigroups and Groups have
Infinite Orbits. Journal of Algebra, 553; 119–137

Gillibert, P. (2018). An Automaton Group with Undecidable Order and Engel Problems. Journal of Algebra, 497; 363-392

Gorenstein, D. and R. Lyons (1983). The Local Structure of Finite Groups of Characteristic 2 Type, Volume 42. American Mathematical Society

Gorenstein, D., R. Lyons, R. Solomon, and M. W. Liebeck (1999). The Classification of The Finite Simple Groups, Number 3. Bulletin of The London Mathematical Society, 31(151); 501–501

Gow, R. (2000). Commutators in Finite Simple Groups of Lie Type. Bulletin of The London Mathematical Society, 32(3); 311–315

Hall Jr, M. (1967). On The Number of Sylow Subgroups in a Finite Group. Journal of Algebra, 7(3); 363–371

Kurdachenko, L., P. Longobardi, and M. Maj (2020a). Groups with Finitely Many Isomorphism Classes of Non-Normal Subgroups. Advances in Group Theory and Applications, 10; 9–41

Kurdachenko, L., A. Pypka, and I. Y. Subbotin (2019a). On The Structure of Groups whose Non-Normal Subgroups are Core-Free. Mediterranean Journal of Mathematics, 16(6); 1–11

Kurdachenko, L., A. Pypka, and I. Y. Subbotin (2020b). On Groups, whose Non-Normal Subgroups are Either Contranormal or Core-Free. Advances in Group Theory and Applications, 10; 83–125

Kurdachenko, L., N. Semko, and I. Y. Subbotin (2020c). Applying Group Theory Philosophy to Leibniz Algebras: Some New Developments. Advances in Group Theory and Applications, 9; 71–121

Kurdachenko, L. A., J. Otal, and I. Y. Subbotin (2019b). On Some Properties of The Upper Central Series in Leibniz Algebras. Commentationes Mathematicae Universitatis Carolinae, 60(2); 161–175

Mannan, M. A., H. Akter, and S. Mondal (2021). Evaluate All Possible Subgroups of a Group of Order 30 and 42 By Using Sylow’s Theorem. International Journal of Scientific & Engineering Research, 12(1); 139–153

McCann, B. (2018). On Products of Cyclic and Abelian Finite p-groups. Proceedings of The Japan Academy, Series A, Mathematical Sciences, 94(8); 77–80

McKay, J. and D. Wales (1971). The Multipliers of The Simple Groups of Order 604, 800 and 50, 232, 960. Journal of Algebra, 17(2); 262–272

Moretó, A. (2021). Multiplicities of Fields of Values of Irreducible Characters of Finite Groups. Proceedings of The American Mathematical Society, 149(10); 4109–4116

Moscatiello, A. L. (2020). Generation of Finite Groups and Maximal Subgroup Growth. Advances in Group Theory and Applications, (2020); 39–49

Robinson, D. J. (2012). A Course in The Theory of Groups, Volume 80. Springer Science & Business Media

Russo, A. (2019). Some Theorems of Fitting Type. Advances in Group Theory and Applications, 8; 39–47

Subbotin, F. d. G. Y. (2019). Some Topics of Classical Group Theory: The Genesis and Current Stage. Advances in Group Theory and Applications, 8; 119–153

Trefethen, S. (2019). Non-Abelian Composition Factors of Finite Groups with The CUT-Property. Journal of Algebra, 522; 236–242

## Authors

Mannan, M. A., Akter, H. ., & Ullah, . M. A. . (2022). 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), 333–343. https://doi.org/10.26554/sti.2022.7.3.333-343