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On non-closure degree for finite groups | ||
| Journal of Algebra and Related Topics | ||
| دوره 13، شماره 2، اسفند 2025، صفحه 109-117 اصل مقاله (159.61 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2025.26910.1639 | ||
| نویسندگان | ||
| A. Suleiman* 1؛ N. M. Mohd Ali2؛ A. I. Kiri3 | ||
| 1Department of Mathematics, Air Force Institute of Technology, Kaduna, Nigeria | ||
| 2Department of Mathematical Sciences, Universiti Teknologi Malaysia, Johor, Malaysia | ||
| 3Department of Mathematical Sciences, Bayero University, Kano, Nigeria | ||
| چکیده | ||
| The non-closure degree for a finite group $G$ has to do with obtaining a probability of selecting any pair of elements $x, y \in G$ such that $xy \notin H$, where $H$ is normal in $G$. It is shown in the paper that the probability lies in the interval $[ 0 , \frac{|G| - 1}{|G|}$ ), with the result as 0 if and only if $H=G$. Illustrations were done using both abelian and non-abelian groups relative to their normal subgroups. | ||
| کلیدواژهها | ||
| Finite group؛ Normal Subgroup؛ closure property of a group؛ Quotient Group | ||
| مراجع | ||
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[1] M. Abdul Hamid, N. M. Muhd Ali, N. H. Sarmin and A. Erfanian, The productivity degree of two subroups of dihedral groups, AIP Conference Proceedings 1605,601-604, doi:10.1063/1.4887657.(ISI) (2014). [2] M. Abdul Hamid, N. M. Muhd Ali, N. H. Sarmin, A. Erfanian and F. N. Abd Manaf, The squared commutativity degree of dihedral groups, Jurnal Teknologi, 78 (2022), 45–49. [3] H. Dubose-Schmidt, M. D. Gallay and D. L. Wilson, Counting nilpotent pairs in finite groups: some conjectures, Mathematical Sciences Technical Reports, 132, (1992). [4] A. Erfanian, B. Toule and N. H. Sarmin, Some considerations on the n-th commutativity degrees of finite groups, Ars Comb., 3 (2011), 495–506. [5] P. Erdos and P. Turan, On some problems of a statistical group theory IV, Acta Mathematica Academiae Scientiarum Hungaricae, 19 (1968), 413–435. [6] W. H. Gustafson, What is the probabilitiy that two group elements commute? The American Mathematical Monthly, (9) 80 (1973), 1031–1034. [7] G. A. Miller, Relative number of non-invariant operators in a group, Proceedings of the National Academy of Sciences, (2) 30 (1944), 25–28. [8] D. M. Patric, C. A. Sugar, G. J. Sherman and E. K. Wespic, What is the probability of generating a cyclic subgroup, Irish Mathematical Society Bulletin, 31 (1993), 22–27. [9] A. I. kiri and A. Suleiman, On independence polynomials of quotient based graphs for finite abelian groups, Journal of the Nigerian Mathematical Society, (3) 41 (2022), 313–323. | ||
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