A Survey of Group Theory

The idea of group theory is to begin with a few of the properties the we all know from basic algebra.  We then see what we can deduce from just these properties.  Then we add more properties and notice what additional facts can then be concluded.  We also look at familiar math structures to see whether they have the given properties.  Now for the definition of a group.

Definition

A Group G is a set with an operator * such that for any a, b, and c are in G then

  1. a*b is in G  (closure)
  2. a*(b*c) = (a*b)*c  (associativity)
  3. There exists an element e of G called the identity element with a*e = e*a = a
  4. There exists an inverse element, a-1 with a-1*a = a*a-1 = e

If the additional property a*b = b*a holds then the group is called an abelian group.

Examples

  • The real numbers, integers, or rational numbers with operator * defined by addition is a group.
  • The real numbers, integers, or rational numbers excluding 0 with operator * defined by multiplication is a group.  Notice that we need to exclude 0 since there is no multiplicative inverse for 0.
  • The set of nxn invertable matrices with operator * defined by matrix multiplication is a group.  Notice that this is not an abelian group, since AB is not necessarily the same as BA.
  • Consider the set {-1,1} where * is defined as multiplication.  This is a group. 

The first three examples are infinite groups, since there are an infinite number of distinct elements of the group.  The fourth example is an example of a finite group.  We say that the order of this group is 2, since there are 2 elements of the group.  In general, for a finite group the order of a group is the number of distinct elements of the group.

 Now for some properties

Property 1

Let G be a group.  Then the identity element is unique.

Proof

First note what it is that we need to prove.  We need to show that if e and f are both elements in G such that for any a in G

a*e = e*a = a     and     a*f = f*a  =  a

Since f is in G, the first set of inequalities take the form

f*e = e*f = f

Since e is in G, the second set of inequalities take the form

e*f = f*e = e

Putting these together gives that e = f and we are done.

Property 2

If

a*x = a*y

then

x = y

Proof

We have

x = e*x = (a-1*a)*x = a-1*(a*x) = a-1*(a*y) = (a-1*a)*y = e*y = y

 

We call this left cancellation.  Notice that a similar argument will show right cancellation.  Also if a*x = x*b then it does not always hold that a = b.

Property 3

The inverse of an element a is unique.

Proof

We need to show that for a in the group, if

x*a = a*x = e

and if

y*a = a*y = e

then x = y.

This follows immediately from Property 2, since

a*x = e = a*y

as we can use right cancellation.

Property 4

(a-1)-1 = a

Proof

We need to show that the inverse of a-1 is a, that is, we need to show that

a*a-1 = a-1*a = e

This is true from the definition of inverse.

Property 5

(a*b)-1 = b-1*a-1

Proof

We have

(a*b)*(b-1*a-1) = a*(b*(b-1*a-1)) = a*((b*b-1)*a-1) = a*(e*a-1) = a*a-1 = e

We leave it to you to show that

(b-1*a-1)*(a*b) = e

Now for examples. 

Example

There is only one group of order 2.  One of its elements must be the identity, e.  Call the other element a.  Since a has an inverse and

a*e = a

the inverse cannot be e.  Therefore it is a.

We have the multiplication table

  e a
a a e
e e a

Notice that this is an abelian group.  When we have a finite group, listing the multiplication table completely defines the group.

Exercises

  1. Determine whether the following are groups.
    1. G is the set of all integers with a*b defined by a - b.
    2. G is the set of all positive integers with a*b defined by multiplication.
    3. G is the set of all reduced rational numbers with odd denominators and a*b is defined by a+b.
    4. G is the set of numbers from 1 to 12  where a*b is defined by clock addition.  For example 8*9 = 17 = 5 (o'clock).
  2. If G is a group such that (a*b)2 = a2*b2 for all a and b in G, show that G is an abelian group.
  3. Let G be a group of order 3.  Show the multiplication table for this group.
  4. Come up with the two multiplication tables for two groups of order 4 such that the first group has an element a with neither a2 nor a3 equal to the identity and the second group has the property that every element is its own inverse.
  5. Come up with the multiplication table for a group of order 6 that is not abelian.  Hint:  consider the group of one-to-one functions on the numbers 1, 2, and 3. 
  6. If G is a finite group, show that there exists a positive integer N such that for all a in G, aN = e.
  7. If G is an abelian group then for all a and b in G and all integers n, (a*b)n = an*bn.  Hint:  Use mathematical induction.
  8. Let G be a group of even order.  Prove that G has at least one element a not equal to the identity such that a2 = e.  Hint:  consider all pairs of the form {a,a-1}.