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Enumerating group actions of finite cyclic groups on finite sets | ||
| Journal of Hyperstructures | ||
| دوره 15، شماره 1، شهریور 2026، صفحه 43-48 اصل مقاله (1.62 M) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22098/jhs.2026.17929.1117 | ||
| نویسنده | ||
| Arezoo Hosseini* | ||
| Department of Mathematics Education Farhangian University | ||
| چکیده | ||
| This paper provides a systematic enumeration of actions of a finite cyclic group Z/nZ on a finite set of size k. We prove that the total number of such actions equals the number of permutations in the symmetric group Sk whose order divides n, and we give an explicit combinatorial sum over cycle type vectors. Furthermore, we show that the number of isomorphism classes of such actions equals the number pD(n)(k) of integer partitions of k into parts drawn from the divisor set D(n) of n. The results connect elementary number theory, partition theory, and permutation group combinatorics. We also briefly discuss extensions to free groups and elementary abelian p-groups, identifying obstacles. | ||
| کلیدواژهها | ||
| Group Actions؛ Finite Cyclic Groups؛ orbit-stabilizer؛ isomorphism classes | ||
| مراجع | ||
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[1] M. Artin. Algebra. Pearson, 2nd edition, 2011. [2] G. E. Andrews. The Theory of Partitions. Cambridge University Press, 1998. [3] W. Burnside. Theory of Groups of Finite Order. Cambridge University Press, 2nd edition, 1911. [4] P. J. Cameron. Oligomorphic permutation groups. In Perspectives in Mathematical Sciences II, 8 (1990), 37–61. [5] P. J. Cameron. Notes on Counting: An Introduction to Enumerative Combinatorics, 26 of Cambridge Texts in Applied Mathematics. Cambridge University Press, 2017. [6] P. Diaconis, J. Fulman, and R. Guralnick. On fixed points of permutations. Journal of Algebraic Combinatorics, 28(1) (2008), 189–218. [7] D. F. Holt, B. Eick, and E. A. O’Brien. Handbook of Computational Group Theory. Chapman & Hall/CRC, 2005. [8] D. S Kumar , P. S Ranjini , H. M R and L. V,. Computation of NM-polnomial and topological indices for cycle related graphs. Journal of Hyperstructures. 14 2 (2025), 242–255, doi: 10.22098/jhs.2025.17644.1100 [9] R. C. Lyndon and P. E. Schupp. Combinatorial Group Theory. Springer, 2001. [10] G.-C. Rota. The number of partitions of a set. The American Mathematical Monthly, 71(5), (1964), 498–504. [11] J. J. Rotman. Advanced Modern Algebra. American Mathematical Society, 2nd edition, 2010. [12] R. P. Stanley. Enumerative Combinatorics, volume 1. Cambridge University Press, 2nd edition, 2011. | ||
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آمار تعداد مشاهده مقاله: 234 تعداد دریافت فایل اصل مقاله: 100 |
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