Sonlu Gruplarda Komütatiflik, Sürjektiflik ve Homomorfizma Dereceleri Arasındaki Nicel İlişkiler

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

Supporting Institution

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

Ethical Statement

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

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

Supporting Institution

This research received no specific grant from any funding agency.

Ethical Statement

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

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