Realización de conceptos abstractos de la teoría de grupos por medio de un modelo computacional: descomposición cíclica de grupos de enteros multiplicativos y visualización de propiedades para grupos no abelianos D4 y Q8.

Abstract

Several of the most important results and applications of group theory underlie finite groups, for this reason there is a singular interest in these structures and the study and exploration of their properties is still in force. Finite groups are subdivided into abelian and non-abelian groups. In the present work this distinction will be taken, and each one will be studied separately, through a computational model programmed in C++ language. For finite abelian groups we will work with one of the most important lemmas of the theory, attributed to the mathematician Kronecker in 1870, known as the Fundamental Theorem of Finite Abelian Groups, which states that each of these groups can be decomposed by cyclic groups. Based on the fundamental theorem, the model will have a function that obtains the cyclic decomposition of the groups of multiplicative integers modulo n. For non-abelian groups, due to the lack of a general result that allows characterizing them, two particular groups have been chosen in order to observe their behavior and draw conclusions that expose the contrasts with the abelian groups; for example, the normal subgroup concept can be better appreciated in non-abelian groups since it is trivial in the abelian case. With this approach, the model also includes implementations corresponding to the groups D4 and Q8, and various concepts such as isomorphism or normal subgroup. All of the above is consolidated in a program that interacts with the user and obtains results according to the model made, and the available options chosen.