Group

Consider a thing with some symmetry. A symmetry is a transformation of a thing which leaves its essential properties unchanged.

A group is all the symmetries of some thing, considered together as a unit.

For any given thing, there are some subtleties in specifying what we mean by its symmetry. Precisely which properties do we consider essential? The answer to that should be covered in the symmetry article.

For groups, there is an additional question we must ask. For any given thing with symmetry, by what standard do we say that two such transformations are "the same" transformation? For example, circles have a rotational symmetry. Let $T_{1}$ denote the transformation which rotates the circle to the left by 90˚ over the course of 1 second, and let $T_{2}$ denote the transfomation which does the same thing over the course of 2 seconds. Are $T_{1}$ and $T_{2}$ to be considered the same transformation, or not?

We will answer this question clearly for discrete symmetries. For continuous symmetries, it is much more subtle and involves some advanced topics.

Symmetries of discrete things

[TODO elaborate]

The idea is that for a discrete symmetry, you can model the "essential" structure that you care about as a finite set, with some structure (things like functions or relations on the set). A symmetry of such a thing is a bijection from the underlying set to itself, such that when you transport the structure across the bijection, you get a structure which is equal to the original structure. It is obvious that the set of all such symmetries is finite, and that it forms a group.

For example, the symmetries of a square are in the following sense the group $D_{4}$. We model a square by a set of vertices $V=\{v_{1},v_{2},v_{3},v_{4}\}$, and a set of relations

$$A=\{\{v_{1},v_{2}\},\{v_{2},v_{3}\},\{v_{3},v_{4}\},\{v_{1},v_{4}\}\}\subset 2^{A}$$

signifying that two vertices are connected by an edge. A symmetry of the square is a bijection $\varphi :V\rightarrow V$ such that if $\{x,y\}\in A$ then $\{\varphi (x),\varphi (y)\}\in A$.

Symmetries of continuous things

To talk about the symmetries of continuous things, you need to get some sort of handle on the continuum. This involves things like Lie theory, algebraic topology, measure theory, and more. It is very advanced, so I don't plan to write about it for a long time. I will, however, give one example that shows the complexity involved.

Consider two pennies sitting on a table. Let's say that the symmetry they possess is that they are identical, so we can swap them. By one way of talking about equivalence, we can say that two swapping transformations are equivalent if and only if they leave the pennies in the same place when they are finished; by that standard the pennies admit a $\mathbb {Z} _{2}$ symmetry. According to another way of talking about equivalence we can say that two swapping transformations are equivalent if and only if the world-lines of the pennies during one transformation are homotopic to the world-lines of the pennies during the other; by that standard the pennies admit a $\mathbb {Z}$ symmetry.