Polmer grupe ${\rm SL}_2({\rm F}_p)$ : delo diplomskega seminarja

Cerar, Matej (author), Jezernik, Urban (mentor)

V delu omejimo polmer Cayleyjevega grafa grupe ${\rm SL}_2({\rm F}_p)$. V ta namen predstavimo Helfgottov produktni izrek. Obravnavamo nekaj robnih primerov za zelo majhne in zelo velike generirajoče podmnožice grupe ${\rm SL}_2({\rm F}_p)$. Definiramo maksimalne toruse in regularne polenostavne elemente, preko katerih zapišemo izrek o torusni dihotomiji. Uporabimo tudi izrek o Larsen-Pinkovih neenakostih in si podrobneje ogledamo konjugiranostne razrede grupe ${\rm SL}_2({\rm F}_p)$.

• Language: Slovenian
• Keywords: Helfgottov produktni izrek, torusi, regularni polenostavni elementi
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2022
• PID: 20.500.12556/RUL-140815
• UDC: 512
• COBISS.SI-ID: 122434051
• Publication date in RUL: 18.09.2022

Secondary language

• Language: English
• Title: Diameter of the group ${\rm SL}_2({\rm F}_p)$

Abstract

In this work, we give an upper bound for the diameter of the Cayley graph of the group ${\rm SL}_2({\rm F}_p)$. For this purpose, we introduce Helfgott's product theorem. We consider some special cases for very small and very large generating subsets of the group ${\rm SL}_2({\rm F}_p)$. We define maximal tori and regular semisimple elements, through which we write down the torus dichotomy theorem. We also present the theorem of Larsen-Pink inequalities and take a closer look at the conjugacy classes of the group ${\rm SL}_2({\rm F}_p)$.

• Keywords: Helfgott's product theorem, tori, regular semisimple elements