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.

Let f be a symmetry (say of a set S in the plane). Is the inverse map f

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:

- S has an
*identity*element e such that for any r∈S (element r in G) e•r=r•e=r. - For any element r∈G, there exists an element
r
^{-1}∈G such that r•r^{-1}=e, the identity element. - For any three elements r,s,t∈G, (r•s)•t=r•(s•t).

Note that the identity element is unique: if G is a group and both
e_{1} and e_{2} are identity elements then we have
e_{1}=e_{1}•e_{2}=e_{2}, by the
properties of identity (condition 1).

- Are the integers with the operation of multiplication and identity 1 a group? (Hint: is there always an inverse?)
- Are the rational numbers with the operation of multiplication and identity 1 a group?
- Are the non-zero real numbers with multiplication a group? What is the identity element?
- Do isometries, with the operation of composition, satisfy the conditions for being a group?
- Do symmetries? (Hint: look back to activity 1. Do symmetries satisfy condition 3?)

- Show that for any group G and g,h∈G,
(g•h)
^{-1}=h^{-1}•g^{-1}. - 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.) - Show that inverses are unique. That is, if g∈G, and there are
two elements h
_{1},h_{2}∈G such that g•h_{1}=e and g•h_{2}=e, then h_{1}=h_{2}. - 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 g^{n}. Similarly,
g^{-1}...g^{-1} n times we write g^{-n} so that
g^{0}=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.

An element g∈G such that g

- 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?
- 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?)
- 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 D_{n}, 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
D_{2n}, since it has 2n elements, others denote it
D_{n}, since it corresponds to an n-gon. Therefore if you see a
D_{3} mentioned, you'll immediately know that the second
convention is being followed, while if you see a D_{6} you must
guess from the context whether it's the symmetry group of a 3-gon or a
6-gon.

Let's look more closely at the dihedral group D_{4}. For
ease of notation, lets denote the identity element by I, the
counterclockwise rotations by R_{90}, R_{180}, and
R_{270}, and the reflections by the names of the corresponding
lines, as in the picture to the right.

Note that
R_{90}L_{1}=L_{2} (recall that
R_{90}L_{1} means reflect by L_{1} and then
rotate 90 degrees counterclockwise) while
L_{1}R_{90}=L_{4}. 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 D_{4} as we've
seen above, is called *non-abelian*.

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 D

- In the table you completed above each row and each column contain every element of the group once. Why?
- The upper left quarter of the table and the lower right quarter only contain rotations. Why?

Next: Generators and Cayley graphs