This
paper aims at treating a study on the order of every element for addition and
multiplication composition in the higher order of groups for different
algebraic structures as groups; order of a group and order of 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 addition and multiplication composition. If G is a finite group, n is
a positive integer anda ⋴ G, then the order of the products na. When G is a finite group, every element must have finite order. However,
the converse is false: there are infinite groups where each element has finite
order. For example, in the group of all roots of unity in C× each
element has finite order. Finally, we find out the order of every element of a
group in different types of higher order of group.
References
[1]
Beltrán, A., Lyons, R., Moretó, A., Navarro, G., Sáez, A. and Tiep, P.H. (2018) Order of Products of Elements in Finite Groups. Journal of the London Mathematical Society, 99, 535-552. https://doi.org/10.1112/jlms.12185
[2]
Gow, R. (2000) Commutators in Finite Simple Groups of Lie Type. Bulletin of the London Mathematical Society, 32, 311-315. https://doi.org/10.1112/S0024609300007141
[3]
Gorenstein, D. and Lyons, R. (1983) The Local Structure of Finite Groups of Characteristic 2 Type. Memoirs of the American Mathematical Society, Vol. 42, American Mathematical Society, Providence. https://doi.org/10.1090/memo/0276
[4]
Gorenstein, D., Lyons, R. and Solomon, R. (1998) The Classification of the Finite Simple Groups, Number 3, Part 3. Mathematical Surveys and Monographs, Vol. 40, American Mathematical Society, Providence. https://doi.org/10.1090/surv/040.3
[5]
Robinson, D.J.S. (1982) A Course in the Theory of Groups. Springer-Verlag, New York.
[6]
Abdul Mannan, Md., Akter, H. and Mondal, S. (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, 139-153
[7]
Hall Jr., M. (1961) On the Number of Sylow Subgroups in a Finite Group. Journal of Algebra, 7, 363-371. https://doi.org/10.1016/0021-8693(67)90076-2
[8]
Brauer, R. (1942) On Groups Whose Order Contains a Prime Number to the First Power I. American Journal of Mathematics, 64, 401-420. https://doi.org/10.2307/2371693
[9]
Mckay, J. and Wales, D. (1971) The Multipliers of the Simple Groups of Order 604,800 and 50,232,960. Journal of Algebra, 17, 262-273. https://doi.org/10.1016/0021-8693(71)90033-0
[10]
Trefethen, S. (2019) Non-Abelian Composition Factors Finite Groups with the CUT-Property. Journal of Algebra, 522, 236-242. https://doi.org/10.1016/j.jalgebra.2018.12.002
[11]
Moretó, A. (2021) Multiplicities of Fields of Values of Irreducible Characters of Finite Groups. Proceedings of the American Mathematical Society, 149, 4109-4116. https://doi.org/10.1090/proc/15566
[12]
Bächle, A. (2019) 3 Questions on Cut Groups. Advances in Group Theory and Applications, 8, 157-160.
[13]
Kurdachenko, L.A., Pypka, A.A. and Subbotin, I.Ya. (2019) On the Structure of Groups Whose Non-Normal Subgroups Are Core-Free. Mediterranean Journal of Mathematics, 16, Article No. 136. https://doi.org/10.1007/s00009-019-1427-6
[14]
Kurdachenko, L.A., Pypka, A.A. and Subbotin, I.Ya. (2020) On Groups Whose Non-Normal Subgroups Are Either Contranormal or Core-Free. Advances in Group Theory and Applications, 10, 83-125.
[15]
Conway, J., Curtis, R., Norton, S., Parker, R. and Wilson, R. (1985) Atlas of Finite Groups. Oxford University, Eynsham.
[16]
Kurdachenko, L.A., Longobardi, P. and Maj, M. (2020) Groups with Finitely Many Isomorphism Classes of Non-Normal Subgroups. Advances in Group Theory and Applications, 10, 9-41.
[17]
McCann, B. (2020) On Products of Cyclic and Non-Abelian Finite p-Groups. Advances in Group Theory and Applications, 9, 5-37.
[18]
Lucchini, A. and Moscatiello, M. (2020) A Probabilistic Version of a Theorem of Laszlo Kovacs and Hyo-Seob Sim. International Journal of Group Theory, 9, 1-6.
[19]
Lucchini, A. and Moscatiello, M. (2020) Generation of Finite Groups and Maximal Subgroup Growth. Advances in Group Theory and Applications, 9, 39-49.
[20]
Cook, W.J., Hall, J., Klima, V.W. and Murray, C. (2019) Leibniz Algebras with Low-Dimensional maximal Lie Quotients. Involve, 12, 839-853. https://doi.org/10.2140/involve.2019.12.839
[21]
Kurdachenko, L.A., Otal, J. and Subbotin, I.Ya. (2019) On Some Properties of the Upper Central Series in Leibniz Algebras. Commentationes Mathematicae Universitatis Carolinae, 60, 161-175. https://doi.org/10.14712/1213-7243.2019.009
[22]
Kurdachenko, L.A., Semko, N.N. and Subbotin, I.Ya. (2020) Applying Group Theory Philosophy to Leibniz Algebras: Some New Developments. Advances in Group Theory and Applications, 9, 71-121.
[23]
Bächle, A. and Margolis, L. (2019) On the Prime Graph Question for Integral Group Rings of 4-Primary Groups II. Algebras and Representation Theory, 22, 437-457. https://doi.org/10.1007/s10468-018-9776-6
[24]
Bächle, A. and Margolis, L. (2019) An Application of Blocks to Torsion Units in Group Rings. Proceedings of the American Mathematical Society, 147, 4221-4231.
[25]
Margolis, L. and del Río, á. (2019) Finite Subgroups of Group Rings: A Survey. Advances in Group Theory and Applications, 8, 1-37.
[26]
Russo, A. (2019) Some Theorems of Fitting Type. Advances in Group Theory and Applications, 8, 39-47.
[27]
de Giovanni, F. and Subbotin, I.Ya. (2019) Some Topics of Classical Group Theory: The Genesis and Current Stage. Advances in Group Theory and Applications, 8, 119-153.
[28]
D’Angeli, D., Francoeur, D., Rodaro, E. and Wächter, J.P. (2020) Infinite Automaton Semigroups and Groups Have Infinite Orbits. Journal of Algebra, 553, 119-137. https://doi.org/10.1016/j.jalgebra.2020.02.014
[29]
Gillibert, P. (2018) An Automaton Group with Undecidable Order and Engel Problems. Journal of Algebra, 497, 363-392. https://doi.org/10.1016/j.jalgebra.2017.11.049
[30]
Cavaleri, M., D’Angeli, D., Rodaro, A.D.E. (2021) Graph Automaton Groups. Advances in Group Theory and Applications, 11, 75-112.
[31]
Casolo, C., Dardano, U. and Rinauro, S. (2018) Groups in Which Each Subgroup Is Commen Surable with a Normal Subgroup. Journal of Algebra, 496, 48-60. https://doi.org/10.1016/j.jalgebra.2017.11.016
[32]
Dardano, U. and Rinauro, S. (2019) Groups with Many Subgroups which Are Commensurable with some Normal Subgroup. Advances in Group Theory and Applications, 7, 3-13.
[33]
Margolis, L. and Schnabel, O. (2019) The Herzog-Schönheim Conjecture for Small Groups and Harmonic Subgroups. Beiträge zur Algebra und Geometrie, 60, 399-418. https://doi.org/10.1007/s13366-018-0419-1
[34]
Chouraqui, F. (2019) The Herzog-Schönheim Conjecture for Finitely Generated Groups. International Journal of Algebra and Computation, 29, 1083-1112. https://doi.org/10.1142/S0218196719500425
[35]
Chouraqui, F. (2018) The Space of Coset Partitions of Fn and Herzog-Schönheim Conjecture. arXiv: 1804.11103.
[36]
Chouraqui, F. (2020) An Approach to the Herzog-Schönheim Conjecture Using Automata. Developments in Language Theory: 24th International Conference, Tampa, 11-15 May 2020, 55-68. https://doi.org/10.1007/978-3-030-48516-0_5
[37]
Chouraqui, F. (2021) Herzog-Schönheim Conjecture, Vanishing Sums of Roots of Unity and Convex Polygons. Communications in Algebra, 49, 4600-4615. https://doi.org/10.1080/00927872.2021.1924766
