Group Theory: A Tutorial

Symmetry means sameness, or indistinguishability, under some transformation. For example, a human face is indistinguishable from its reflected version, and therefore we say that it is reflectionally symmetric. A circle is indistinguishable from any of its rotated versions (about its center), and therefore we say that it is rotationally symmetric. When one puts together all the transformations that can be applied to an object, that leave it indistinguishable from its original version, one says that one has a group of transformations. Standardly, in mathematics, symmetry is characterized by the concept of a group of transformations. Fortunately, the concept of group is one of the simplest to understand in mathematics. One can think of a group simply as a list of the symmetries contained in an object. For example, a cube has a total of 48 symmetries, which are given by its reflectional and rotational axes. The group of a cube is therefore the list of those 48 symmetries. Any object has its own group which is a list of its symmetries. The purpose of this tutorial is to give the reader an introduction to groups and to the way they describe symmetries.