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[4] viXra:2511.0115 [pdf] submitted on 2025-11-23 00:26:46
Authors: Teo Banica
Comments: 400 Pages.
This is an introduction to the finite groups, with focus on the groups of permutations and reflections, and more generally, on the finite groups of unitary matrices. We first discuss the basics of group theory, featuring the cyclic, dihedral and symmetric groups, and the structure result for finite abelian groups. Then we study the complex reflection groups, with general theory and examples, classification results, and with a look into braid groups too. We then go into the study of representation theory, and of more advanced aspects, such as Tannakian duality, Brauer theorems and Clebsch-Gordan rules. Finally, we discuss, using representation theory methods, a number of advanced analytic aspects, for the most in relation with questions coming from probability.
Category: Algebra
[3] viXra:2511.0105 [pdf] replaced on 2025-11-24 14:18:22
Authors: Timothy Jones
Comments: 3 Pages. There was a slight error in the previous version: S4 is not the symmetries of a square.
We quickly define permutation group S3 and then make a Cayley Table for it. We use a surprisingly simple program written on a TI84-CE calculator.
Category: Algebra
[2] viXra:2511.0073 [pdf] replaced on 2025-11-29 02:19:28
Authors: Shuojie Yuan
Comments: 5 Pages.
This paper studies the positive integer solutions of an exponential Diophantine equation of the following form: 3^aX+3^{a-1}+3^{a-2}times2^{b_1}+3^{a-3}times2^{b_1+b_2}ldots+3^{a-a}times2^{b_1+b_2+ldots+b_{a-1}}}{2^{b_1+b_2+ldots+b_a}}=X，where a,X,b_iare positive integers andb_igeq1. This equation arises from the analysis of the cyclic structure of a specific iterative sequence. Through rigorous algebraic derivation and inequality analysis, this paper proves that for any given ageq1 and all sequences of b_i satisfying the conditions, the only positive integer solution to this equation is X=1 . This conclusion reveals the uniqueness of the solution for this class of equations.
Category: Algebra
[1] viXra:2511.0007 [pdf] submitted on 2025-11-02 10:39:56
Authors: Absos Ali Shaikh, Uddhab Roy
Comments: 42 Pages.
The purpose of this expository article is to introduce the notion of a new type of (classical and non-classical) Schottky structures in the discrete subgroups of the projective special linear group over the real numbers of degree $2$. In particular, in this manuscript, we have investigated the classical and non-classical structures of a kind of Schottky group (which we named as semi-Fuchsian Schottky groups) in the hyperbolic plane. In general, a Schottky group is called classical if the Schottky curves used in the Schottky construction are Euclidean circles; on the other hand, it is said to be non-classical if the defining curves are Jordan curves, except the Euclidean circles. In fact, in this article, we initiated the concept of classical semi-Fuchsian Schottky groups of rank $2$ (hence any finite rank) in the upper-half plane model. This study yields new and interesting surfaces in Riemann surface theory, specifically, various types of hyperbolic pairs of pants with infinite area (indeed, we proposed the notion of a pair of full pants) and a hyperbolic torus with one infinite end. After that, we constructed a structure of the rank $2$ semi-Fuchsian Schottky group with non-classical generating sets in the hyperbolic plane by using two suitable M"obius transformations. More precisely, in this paper, we have produced a non-trivial example of the semi-Fuchsian Schottky group of rank $2$ with non-classical generating sets within the hyperbolic plane.
Category: Algebra
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