Sonlu Gruplarda Komütatiflik, Sürjektiflik ve Homomorfizma Dereceleri Arasındaki Nicel İlişkiler

Year 2025, Volume: 29 Issue: 3, 705 - 715, 25.12.2025

Mehmet Uc

https://doi.org/10.19113/sdufenbed.1818549

Abstract

Bu makale, sonlu gruplarda yapısal analiz amacıyla iki yeni olasılıksal ölçüyü, sür-jektiflik derecesi ve homomorfizm derecesini tanıtmaktadır. Bu ölçütler ile daha önce tanıtılan komütatiflik derecesi arasında ilişki kuran analitik bir çerçeve geliştirilmiştir. Sürjektiflik derecesine bağlı olarak, komütatiflik derecesi için yeni alt ve üst sınırlar elde edilmiş ve gruplar arasındaki fonksiyonların homomorfizma özellikleri nicel olarak incelenmiştir. Kavramlar arasındaki ilişkiler teoremler ve örneklerle desteklenmiş olup, bazı örnekler için SageMath kodu verilmiştir. Bu bulgular yapısal homomorfizmlerin daha derinlemesine olasılıksal olarak anlaşıl-masına katkıda bulunmakta ve sonlu gruplar içindeki cebirsel ilişkilerin niceliksel olarak belirlenmesi için yeni analitik araçlar sağlamaktadır.

Keywords

Komütatiflik derecesi , Grup homomorfizması , Surjektiflik

Ethical Statement

Yazar, bu çalışmada herhangi bir etik ihlal bulunmadığını beyan etmektedir.

Supporting Institution

Bu araştırma herhangi bir kurum veya kuruluş tarafından maddi olarak desteklenmemiştir.

Quantitative Relations between Commutativity, Surjectivity, and Homomorphism Degrees in Finite Groups

Abstract

This article introduces two new probabilistic measures, the surjectivity degree and the homomorphism degree, for the purpose of structural analysis in finite groups. An analytical framework is developed that establishes a relationship be-tween these measures and the previously introduced commutativity degree. New lower and upper bounds for the commutativity degree, depending on the surjectiv-ity degree, are obtained; the homomorphism properties of functions between groups are quantitatively investigated. The relationships between the concepts are supported by theorems and examples, and SageMath code is provided for some examples. These findings contribute to a deeper probabilistic understanding of structural homomorphisms and provide new analytical tools for quantifying algebraic relationships within finite groups.

Keywords

Commutativity degree , Group homomorphism , Surjectivity

Ethical Statement

The author declares that there is no ethical issue in this study.

Supporting Institution

This research received no specific grant from any funding agency.

References

• [1] Gallagher, P. X. 1970. The number of conjugacy classes in a finite group. Mathematische Zeitschrift, 118(3), 175–179.
• [2] Gustafson, W. H. 1973. What is the probability that two group elements commute? The Ameri-can Mathematical Monthly, 80(9), 1031–1034.
• [3] Rusin, D. 1979. What is the probability that two elements of a finite group commute? Pacific Journal of Mathematics, 82(1), 237–247.
• [4] Gürdal, M. 2009. Description of extended eigen-values and extended eigenvectors of integration operators on the Wiener algebra. Expositiones Mathematicae, 27(2), 153–160.
• [5] Karaev, M., Gürdal, M., Huban, M. B. 2016. Re-producing kernels, Engliš algebras and some applications. Studia Mathematica, 232(2), 113–141.
• [6] Karaev, M., Gürdal, M., Saltan, S. 2011. Some applications of Banach algebra techniques. Mathematische Nachrichten, 284(13), 1678–1689.
• [7] Gürdal, M., Garayev, M. T., Saltan, S. 2015. Some concrete operators and their properties. Turk-ish Journal of Mathematics, 39(6), 970–989.
• [8] Gürdal, M. 2009. On the extended eigenvalues and extended eigenvectors of shift operator on the Wiener algebra. Applied Mathematics Let-ters, 22(11), 1727–1729.
• [9] Lescot, P. 1995. Isoclinism classes and commu-tativity degrees of finite groups. Journal of Al-gebra, 177(3), 847–869.
• [10] Lescot, P. 2001. Central extensions and commu-tativity degree. Communications in Algebra, 29(10), 4451–4460.
• [11] Moghaddam, M. R. R., Salemkar, A. R., Chiti, K. 2005. n-Isoclinism classes and n-nilpotency degree of finite groups. Algebra Colloquium, 12(2), 255–261.
• [12] Erfanian, A., Rezaei, R., Lescot, P. 2007. On the relative commutativity degree of a subgroup of a finite group. Communications in Algebra, 35(12), 4183–4197.
• [13] Tărnăuceanu, M. 2009. Subgroup commutativi-ty degrees of finite groups. Journal of Algebra, 321(9), 2508–2520.
• [14] Barzegar, R., Erfanian, A., Farrokhi, M. D. G. 2013. Finite groups with three relative commu-tativity degrees. Bulletin of the Iranian Mathe-matical Society, 39(2), 271–280.
• [15] Rezaei, R., Erfanian, A. 2014. A note on the relative commutativity degree of finite groups. Asian-European Journal of Mathematics, 7(1), 1450017.
• [16] Pournaki, M. R., Sobhani, R. 2008. Probability that the commutator of two group elements is equal to a given element. Journal of Pure and Applied Algebra, 212(4), 727–734.
• [17] Nath, R. K., Das, A. K. 2010. On a lower bound of commutativity degree. Rendiconti del Circolo Matematico di Palermo, 59(1), 137–142.
• [18] Pirzadeh, M., Hashemi, M. 2022. A generaliza-tion of the n^th-commutativity degree in finite groups. Computational Sciences and Engineer-ing, 2(1), 33–40.
• [19] Ghaneei, M., Azadi, M. 2021. The nth commuta-tivity degree of semigroups. Journal of Linear and Topological Algebra, 10(3), 225–233.
• [20] Nath, R. K., Das, A. K. 2011. On generalized commutativity degree of a finite group. The Rocky Mountain Journal of Mathematics, 41(6), 1987–2000.
• [21] Chashiani, A., Rezaei, R. 2021. On the commuta-tivity degree of a group algebra. Afrika Ma-tematika, 32(5), 1137–1145.
• [22] Arvasi, Z., Çağlayan, E. I., Odabaş, A. 2022. Commutativity degree of crossed modules. Turkish Journal of Mathematics, 46(1), 242–256.
• [23] Çetin, S., Gürdal, U. 2024. Crossed modules with action. Ukrainian Mathematical Journal, 76(4), 649–668.