A method for data analysis in algebraic structures

Abstract

Computational group theory investigates the arithmetic properties of algebraic structures. These properties may be utilized in development of algorithms for data analysis and machine learning. This direction is associated with the design, analysis of algorithms and data structures for calculating various characteristics for (mostly finite) groups. The area is interesting for investigation of important from various points of view groups that are important, the data about which cannot be obtained by manually. The present work was carried out in line with computer calculations in groups. Based on the concept of the group growth function, the concept of the group density function is introduced. The growth and density functions of finite simple non-Abelian groups of small orders are constructed. The question of recognizability of finite simple non-Abelian groups by the density function is considered. The problem of effective storage of finite group elements in computer memory using the AVL tree is investigated. Based on the AVL tree, a fast algorithm for enumerating all elements of a finite group is developed. With the computer calculations, the recognizability of two finite simple non-Abelian groups by the group density function is proved.

Regular permutations and their applications in crystallography

“…The concepts mentioned below, namely permutations, regular permutations, permutation degrees, and permutation orders, are widely known. Representing a group G in the form of regular permutations is extensively used to investigate the structure of finite groups, especially for parameters such as group density function, group spectrum, and the Cayley graph of the group [8,9,10]. This is related to the increased capabilities of computer technologies for conducting computations.…”