Groups of symmetries

Last time we examined symmetries and isometries as geometric objects: distance preserving maps from a space to itself, in case of symmetries leaving a given set in that space invariant. This time we will focus on their algebraic properties.

Activity 1: Going backwards
Let f be a symmetry (say of a set S in the plane). Is the inverse map f-1 also a symmetry? (Hint: is the inverse of an isometry an isometry? Is every point in S taken by f-1 to a point in S?)

A group is a set G with an operation • which takes two elements of G and outputs one element (like, for example addition or multiplication) that satisfy the following conditions:

  1. S has an identity element e such that for any r∈S (element r in G) e•r=r•e=r.
  2. For any element r∈G, there exists an element r-1∈G such that r•r-1=e, the identity element.
  3. For any three elements r,s,t∈G, (r•s)•t=r•(s•t).
One example of a group is the set of integers with the operation of addition and the identity element 0. Another example is a set with two elements, e and b, such that b•e=e•b=b, b•b=e, and e•e=e.

Note that the identity element is unique: if G is a group and both e1 and e2 are identity elements then we have e1=e1•e2=e2, by the properties of identity (condition 1).

Activity 2: Groups and non-groups
  1. Are the integers with the operation of multiplication and identity 1 a group? (Hint: is there always an inverse?)
  2. Are the rational numbers with the operation of multiplication and identity 1 a group?
  3. Are the non-zero real numbers with multiplication a group? What is the identity element?
  4. Do isometries, with the operation of composition, satisfy the conditions for being a group?
  5. Do symmetries? (Hint: look back to activity 1. Do symmetries satisfy condition 3?)
Activity 3: Properties of groups
  1. Show that for any group G and g,h∈G, (g•h)-1=h-1•g-1.
  2. Show that if G is a group, then for any g∈G, g-1•g=e. (Hint: show that for any element h∈G, if if h•h=h then h=e and use condition 2.)
  3. Show that inverses are unique. That is, if g∈G, and there are two elements h1,h2∈G such that g•h1=e and g•h2=e, then h1=h2.
  4. Show that for any g∈G, (g-1)-1=g. (Hint: use part 3.)

From now on, for the sake of brevity, we'll omit the operation symbol • and simply write gh instead of g•h. Also, we abbreviate gg...g (n times) as gn. Similarly, g-1...g-1 n times we write g-n so that g0=e.

Above we established that symmetries are indeed groups. One way to build a repertoire of examples of groups is to examine the symmetries of various sets and figures and isometries of various spaces.

Activity 4: An order of symmetries
An element g∈G such that gn=e but gn-1≠e for some integer n>0 is called an element of order n. If gn≠e for every n, we say g has infinite order.
  1. Consider only the rotational symmetries of an n-sided regular polygon (that is, the subset of symmetries which are rotations). Do they constitute a group? For a given n, how many elements does this group have? What orders do these elements have?
  2. Now take the full symmetry group of an n-sided regular polygon. How many elements does it have? How many of those elements have order 2? (Hint: look back at the case of the square, analyzed in the last section. If n is odd, is there an order two rotation?)
  3. Consider the integer points on a real line. What are their symmetries? Are there any symmetries with orders other than 2 or infinity? (Hint: recall that the isometries of the real line are translation and reflections about a point. Which of these map the integers to themselves?)

The group from number 2 above is called the dihedral group on 2n elements, usually denoted Dn, while the group in number 3 is called the infinite dihedral group, denoted D.

Unfortunately there is no consensus among mathematicians (or textbook writers) on how to denote the dihedral group corresponding to the symmetries of an n-gon. Some denote it D2n, since it has 2n elements, others denote it Dn, since it corresponds to an n-gon. Therefore if you see a D3 mentioned, you'll immediately know that the second convention is being followed, while if you see a D6 you must guess from the context whether it's the symmetry group of a 3-gon or a 6-gon.
square and marked lines

Let's look more closely at the dihedral group D4. For ease of notation, lets denote the identity element by I, the counterclockwise rotations by R90, R180, and R270, and the reflections by the names of the corresponding lines, as in the picture to the right.

Note that R90L1=L2 (recall that R90L1 means reflect by L1 and then rotate 90 degrees counterclockwise) while L1R90=L4. Groups G in which any two elements g,h∈G satisfy the equality gh=hg are called abelian. A group where this relationship is not satisfied for at least one pair of elements, like the group D4 as we've seen above, is called non-abelian.

Activity 5: Groups by the table
One way to define a (finite) group is by its group product table. To make such a table simply list the elements of the group across the top and the side of the table and fill in their products in the corresponding cells. Part of the product table for D4 is given below, with the convention of column•row. Fill in the missing entries (the solution is given in the next section).
product table for D4
Activity 6: Two final whys
  1. In the table you completed above each row and each column contain every element of the group once. Why?
  2. The upper left quarter of the table and the lower right quarter only contain rotations. Why?

Next: Generators and Cayley graphs