Transformation groups and symmetry groups

Mathematicians classify the various patterns by their symmetries, the transformations that leave them invariant. For a given pattern, the collection of symmetries form what mathematicians call a symmetry group which is a kind of "transformation group." Now, the word "group" as used in the English language just means a bunch of things considered together. Mathematicians need a word for something more, and, for better or for worse, they decided on "group."

A group of things, for a mathematician, means a collection of things with a certain structure. The structure is one of "composition." Given two elements S and T of a group, you can "compose" them to get another element ST of the group. In our case we're composing transformations of the plane that leave a pattern invariant. That just means first perform one transformation S, then perform the other transformation T. (It's a matter of convention whether you read ST from left to right or from right to left. Although the right-to-left convention is more common, lets use the left-to-right convention here.) If each transformation is a symmetry of a pattern, then their composition is a symmetry of the pattern, too.

The structure, that is, the operation of composition, isn't enough by itself to give a group. There are certain axioms that the operation must obey in order to have a group.

First, there has to be some element of the group that acts as an identity, that is, it doesn't do anything. We'll use the letter I to denote the identity of a group. A little more precisely, the axiom requires that the identity I when composed with any element T gives back T. Algebraically, we require

In a symmetry group for a pattern, the identity is the identity transformation. That's the transformation of the plane that doesn't move any point. It's trivial. It doesn't do anything!

The second requirement for a group is that there are "inverses." An inverse of an element T is another element, usually written T^-1, whose composition with T gives the identity. That is,

So, the inverse undoes whatever T does. For example, the inverse of a translation upwards is a translation downwards. The inverse of a rotation 90° clockwise is a rotation 90° counterclockwise. The inverse of a reflection, surprisingly enough, is itself.

The third axiom for groups is associativity. Algebraically, whenever S, T, and U are three elements,

It allows us to write the composition of three elements without using parentheses. Composition of transformations is associative.

The groups that we're dealing with aren't commutative, that is, the equation

usually doesn't hold in our transformation groups. If both S and T happen to both be translations, then it does hold, but rarely otherwise. For example, if S and T are reflections with parallel axes, then ST and TS are both translations, but in opposite directions.

Up to the table of contents
Back to lattices
On to the 17 plane groups

© 1997.

David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610

The files are located at http://aleph0.clarku.edu/~djoyce/wallpaper/