First introduction to group theory and ring theory - explaining branching structures in an easy-to-understand way

Author: Zen and the Art of Computer Programming

1 Introduction

What is group theory? Why study group theory? Let’s start with an example! "Backgammon" is a classic chess game. In this game, each move consumes five stars, while in Chinese chess, a piece can refer to two spaces or three squares in two columns at the same time. If you want to eliminate five stars at the same time, you need to do the following four things:

1. First, choose a method of elimination. There are two ways to eliminate five stars: kill four in one step and kill four in two steps.
2. Then, depending on the elimination method selected, mark the four or eight spaces next to the drop point or the nine cells in the two columns.
3. Replace the marked four or eight spaces or nine squares in two columns on the board.
4. Depending on the elimination method, choose different chess pieces and continue to fall until they are eliminated.

When we play backgammon, we are actually applying group theory. The components of "backgammon" (five stars) are elements in group theory, and the placement points (spaces) and chess pieces (pieces) of "backgammon" are objects in group theory. During the game, we use some concepts and rules in group theory (for example: generators of groups, bimorphisms) to eliminate "backgammon". Group theory is a discipline used to study the properties of groups and how to use group theory to solve various problems.

Why study group theory? If you are not a natural science or computer science student, you may not have been exposed to group theory, so it will take some time and effort to learn it. However, if you are already an undergraduate or graduate student in an institution of higher learning with a solid foundation, then learning group theory may seem like a miracle. You don't need to memorize those inexplicable symbols, theorems, and formulas by rote, but use the perspective, intuitive experience and systematic learning skills brought by learning group theory to understand the true meaning of group theory. In addition, group theory in modern times