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

I T = T, and T I = T.

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,

T-1 T = I, and T T-1 = I.

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,

(S T) U = S (T U).

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

S T = T S

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.

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Back to lattices
On to the 17 plane groups

© 1997.

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

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