Groups

A Very Special Kind of Set with an Operation: Groups!

In mathematics there is one particular kind of set with an operation that is fundamental to many, many mathematical applications. This special kind of set with an operation is called a group.

 

A group is a set with an operation that has the following 4 properties:

1) The set is closed under the operation.

2) The set is associative under the operation.

3) The set has an identity element under the operation that is also an element of the set.

4) Every element of the set has an inverse under the operation that is also an element of the set.

 

Notice that a group need not be commutative!

 

Letís look at some examples so that we can identify when a set with an operation is a group:

1) The set of integers is a group under the OPERATION of addition:

We have already seen that the integers under the OPERATION of addition are CLOSED, ASSOCIATIVE, have IDENTITY 0, and that any integer x has the INVERSE −x. Because the set of integers under addition satisfies all four group PROPERTIES, it is a group!

 

2) The set {0,1,2} under addition is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the CLOSURE PROPERTY (see the previous lectures to see why). Therefore, the set {0,1,2} under addition is not a group!

(Notice also that this set is ASSOCIATIVE, and has an IDENTITY which is 0, but does not have the INVERSE PROPERTY because −1 and −2 are not in the set!)

 

3)The set of integers under subtraction is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the ASSOCIATIVE PROPERTY (see the previous lectures to see why). Therefore, the set of integers under subtraction is not a group!

(Notice also that this set is CLOSED, but does not have an IDENTITY and therefore also does not have the INVERSE PROPERTY.)

 

4) The set of natural numbers under addition is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the IDENTITY PROPERTY (see the previous lectures to see why). Therefore, the set of natural numbers under addition is not a group!

(Notice also that this set is CLOSED, ASSOCIATIVE, but does not have the INVERSE PROPERTY because none of the negative numbers are in the set.)

 

5) The set of whole numbers under addition is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the INVERSE PROPERTY (see the previous lectures to see why). Therefore, the set of whole numbers under addition is not a group!

(Notice also that this set is CLOSED, ASSOCIATIVE, and has the IDENTITY ELEMENT 0.)

 

6) The set of rational numbers with the element 0 removed is a group under the OPERATION of multiplication:

We have already seen that the set rational numbers with the element 0 removed under the OPERATION of multiplication is CLOSED, ASSOCIATIVE, have IDENTITY 1, and that any integer x has the INVERSE . Because the set of rational numbers with the element 0 removed under multiplication satisfies all four group PROPERTIES, it is a group!

 

7) The set of rational numbers (which contains 0) under multiplication is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the INVERSE PROPERTY (see the previous lectures to see why). Therefore, the set rational numbers under multiplication is not a group!

(Notice also that this set is CLOSED, ASSOCIATIVE, and has an IDENTITY which is 1.)

 

8) The set of rational numbers under division is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the ASSOCIATIVE PROPERTY (see the previous lectures to see why). Therefore, the set of rational numbers under division is not a group!

(Notice also that this set is not CLOSED because anything divided by 0 is not in the set, does not have an IDENTITY and therefore also does not have the INVERSE PROPERTY.)

 

9) The set of natural numbers under division is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the IDENTITY PROPERTY (see the previous lectures to see why). Therefore, the set of natural numbers under division is not a group!

(Notice that this set does not have the CLOSURE, ASSOCIATIVE or INVERSE PROPERTIES.)

 

10) The set of integers under multiplication is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the INVERSE PROPERTY (see the previous lectures to see why). Therefore, the set of integers under multiplication is not a group!

(Notice also that this set is CLOSED, ASSOCIATIVE, and has the IDENTITY ELEMENT 1.)

i) Let’s consider the OPERATION @ acting on the set {β,γ,δ}given by the following group TABLE:

@

β

γ

δ

β

β

δ

δ

γ

δ

β

γ

δ

γ

δ

β

We recall from the last section that this set does not have an IDENTITY ELEMENT under the OPERATION of @ so we know right away that it does not have the INVERSE PROPERTY!

 

j) Here is an operation table for the set {a,b,c} and the OPERATION &:

&

a

b

c

a

b

a

c

b

a

b

c

c

c

c

b

We can see that this set does have the IDENTITY PROPERTY and that the IDENTITY ELEMENT of this set is b.

To find any inverses that might exist, first we look at the table and find any place where the result of an operation on two elements is the IDENTITY element b:

&

a

b

c

a

b

a

c

b

a

b

c

c

c

c

b

From the table we can see that:

a*a= b

b*b= b

c*c= b

So every element has an inverse:

a is the inverse of a

b is the inverse of b

c is the inverse of c

In this set, every element is its own inverse!

So the set {a,b,c} under the operation & defined by the operation table above does have the INVERSE PROPERTY!

 

k) Let’s consider the OPERATION £ acting on the set {β,γ,δ} given by the following OPERATION TABLE:

£

β

γ

δ

β

β

β

β

γ

β

γ

γ

δ

β

γ

δ

We recall from the previous section that this set does have an IDENTITY ELEMENT under the OPERATION of £.

So now let’s try to find any INVERSES that might exist on the table:

 

To find any inverses that might exist, first we look at the table and find any place where the result of an operation on two elements is the IDENTITY element δ:

£

β

γ

δ

β

β

β

β

γ

β

γ

γ

δ

β

γ

δ

From the table we can see that:

δ£δ= δ

So δ, because it is the IDENTITY, is it’s own INVERSE.

But there is no element x so that x£β=δ or β£x=δ, so β does not have an INVERSE!

And there is no element x so that x£γ=δ or γ£x=δ, so γ does not have an INVERSE!

 

We need every element to have an INVERSE in order for the set under the given operation to have the INVERSE PROPERTY!

So the set {β,γ,δ} under the OPERATION £ defined by the operation table above does not have the INVERSE PROPERTY!

 

l) Let’s consider the OPERATION $ acting on the set {1,2,3,4} given by the following OPERATION TABLE:

$

1

2

3

4

1

4

1

2

1

2

1

2

3

4

3

2

3

1

2

4

2

4

3

3

We can see that this set under the operation $ does have an IDENTITY ELEMENT. The IDENTITY ELEMENT is 2.

So now let’s try to find any INVERSES that might exist on the table:

 

To find any inverses that might exist, first we look at the table and find any place where the result of an operation on two elements is the IDENTITY element 2:

$

1

2

3

4

1

4

1

2

1

2

1

2

3

4

3

2

3

1

2

4

2

4

3

3

From the table we can see that:

1$3= 2

2$2= 2

3$4= 2

3$1= 2

4$1= 2

 

To find any INVERSES that might exist, let’s break these down so that we can look at each element of the set one at a time:

 

First let’s consider 2:

2$2= 2

So 2, because it is the IDENTITY, is it’s own INVERSE.

 

Now let’s look at 3:

3$4= 2

1$3= 2 3$1= 2

So while 3$4= 2 , because 4$3≠2, 4 is not an INVERSE of 3.

However, because 1$3= 2 and 3$1= 2 , 1 is an INVERSE of 3!

 

Next let’s consider 1:

4$1= 2

1$3= 2 3$1= 2

So while 4$1= 2 , because 1$4≠2, 4 is not an INVERSE of 1.

However, because 1$3= 2 and 3$1= 2 , 3 is an INVERSE of 1!

 

And finally, let’s look at 4:

3$4= 2

4$1= 2

So because 3≠1, there is no element x so that x^4=2 and 4^x=2. Therefore, 4 does not have an INVERSE!

 

So 1, 2, and 3 have INVERSES in this set under this OPERATION, but 4 does not have an INVERSE.

 

We need every element to have an INVERSE in order for the set under the given operation to have the INVERSE PROPERTY!

So the set {1,2,3,4} under the OPERATION $ defined by the operation table above does not have the INVERSE PROPERTY!

 

Now return to Blackboard to answer Group Lecture Questions 7: Groups!