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
also a symmetry? (Hint: is the inverse of an isometry an
isometry? Is every point in S taken by f-1
to a point
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:
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
- 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
- For any three elements r,s,t∈G,
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
- 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
- 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?)
Activity 3: Properties of groups
- Show that for any group G and g,h∈G,
- 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 h1,h2∈G such that
g•h1=e and g•h2=e, then
- 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
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
element g∈G such that gn
=e but gn-1
some integer n>0 is called an element of order n
≠e for every n, we say g has infinite order.
- 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 Dn, while the
group in number 3 is called the infinite dihedral group, denoted
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
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.
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
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).
Activity 6: Two final whys
- 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