Geometry

Group Theory for Beginners: An Intuitive Approach

What exactly is a symmetry?

A dodecahedron over a Penrose tiling. Images courtesy of Wikimedia Commons.

When we talk about mathematics in terms of its beauty, its symmetry, and its power, we often leave off exactly what these things are. Have you ever stopped to ask yourself what exactly a symmetry is?

In this article, I will describe the mathematical study known as Group Theory, which is the basis for the fields known as Abstract Algebra and Modern Algebra. Don’t be discouraged by the wording, it’s not so difficult after all.

This article is written for anyone with even a basic knowledge of geometry and algebra from high school. It’s ok if you have forgotten it all: we will start from the beginning! We will build the subject from the ground up; so all you need to know is some basic facts about sets and numbers. My main advice in getting started is to not overthink things; nothing is more than what the definition says it is. This is some truly simple mathematics.

Roadmap of the Article:

• Symmetries of the triangle intuitively
• Group definition
• Some facts about groups, and some examples of groups
• Definition of a Geometry, and a Symmetry (!!)

Symmetries of the Triangle