Application of Several Theorems in Number Theory to Group Theory

1905 ◽  
Vol 12 (4) ◽  
pp. 81
Author(s):  
G. A. Miller
Keyword(s):  
Number Theory ◽  
Group Theory

Parametrizing Fuchsian Subgroups of the Bianchi Groups

1991 ◽  
Vol 43 (1) ◽  
pp. 158-181 ◽  
Author(s):  
C. Maclachlan ◽  
A. W. Reid
Keyword(s):  
Number Theory ◽  
Group Theory ◽  
Present Article ◽  
Ring Of Integers ◽  
Low Dimensional Topology ◽  
Low Dimensional

Let dbe a positive square-free integer and let Od denote the ring of integers in . The groups PSL2(Od) are collectively known as the Bianchi groups and have been widely studied from the viewpoints of group theory, number theory and low-dimensional topology. The interest of the present article is in geometric Fuchsian subgroups of the groups PSL2(Od). Clearly PSL2 is such a subgroup; however results of [18], [19] show that the Bianchi groups are rich in Fuchsian subgroups.

On some metabelian 2-groups and applications III

2016 ◽  
Vol 09 (04) ◽  
pp. 1650072
Author(s):  
Abdelkader Zekhnini ◽  
Abdelmalek Azizi ◽  
Mohammed Taous
Keyword(s):  
Number Theory ◽  
Group Theory ◽  
Theoretical Results

The capitulation problem is one of the most important topic in number theory, and as it is closely related to the group theory, we present, in this paper, some group theoretical results to solve this problem, in a particular case, whenever [Formula: see text] for some metabelian [Formula: see text]-group [Formula: see text]. Then we illustrate our results by some examples.

scholarly journals On the Clifford collineation, transform and similarity groups. I.

1961 ◽  
Vol 2 (1) ◽  
pp. 60-79 ◽  
Author(s):  
Beverley Bolt ◽  
T. G. Room ◽  
G. E. Wall
Keyword(s):  
Number Theory ◽  
Group Theory ◽  
Similarity Groups ◽  
Clifford Groups

Papers I, II of this projected series lay the algebraic foundations of the theory of the Clifford groups; I deals with the casep> 2, II with the casep= 2. The present introduction refers to both papers. Our theory has applications in group theory, geometry and number theory.

The Applications of Group Theory

2012 ◽  
Vol 430-432 ◽  
pp. 1265-1268
Author(s):  
Xiao Qiang Guo ◽  
Zheng Jun He
Keyword(s):  
Number Theory ◽  
Algebraic Geometry ◽  
Group Theory ◽  
Algebraic Number ◽  
Algebraic Topology ◽  
Polynomial Equation ◽  
Algebraic Number Theory ◽  
Simple Groups ◽  
Finite Simple Groups

Since the classification of finite simple groups completed last century, the applications of group theory are more and more widely. We first introduce the connection of groups and symmetry. And then we respectively introduce the applications of group theory in polynomial equation, algebraic topology, algebraic geometry , cryptography, algebraic number theory, physics and chemistry.

Preliminaries

2020 ◽  
pp. 5-56
Author(s):  
Andreas Bolfing
Keyword(s):  
Field Theory ◽  
Number Theory ◽  
Group Theory ◽  
Complexity Theory ◽  
Prime Numbers ◽  
Ring Theory ◽  
Cryptographic Primitives ◽  
Mathematical Concepts ◽  
Cyclic Groups ◽  
Efficiency Of Algorithms

Blockchains are heavily based on mathematical concepts, in particular on algebraic structures. This chapter starts with an introduction to the main aspects in number theory, such as the divisibility of integers, prime numbers and Euler’s totient function. Based on these basics, it follows a very detailed introduction to modern algebra, including group theory, ring theory and field theory. The algebraic main results are then applied to describe the structure of cyclic groups and finite fields, which are needed to construct cryptographic primitives. The chapter closes with an introduction to complexity theory, examining the efficiency of algorithms.

scholarly journals PRIMITIVE PRIME DIVISORS AND THE TH CYCLOTOMIC POLYNOMIAL

2015 ◽  
Vol 102 (1) ◽  
pp. 122-135 ◽  
Author(s):  
S. P. GLASBY ◽  
FRANK LÜBECK ◽  
ALICE C. NIEMEYER ◽  
CHERYL E. PRAEGER
Keyword(s):  
Number Theory ◽  
Group Theory ◽  
Cyclotomic Polynomial ◽  
Linear Groups ◽  
Prime Divisors ◽  
Definition Of ◽  
Finite Linear

Primitive prime divisors play an important role in group theory and number theory. We study a certain number-theoretic quantity, called $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)$, which is closely related to the cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{n}(x)$ and to primitive prime divisors of $q^{n}-1$. Our definition of $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)$ is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants $c$ and $k$, we provide an algorithm for determining all pairs $(n,q)$ with $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)\leq cn^{k}$. This algorithm is used to extend (and correct) a result of Hering and is useful for classifying certain families of subgroups of finite linear groups.

Sign in / Sign up

Export Citation Format

Share Document