Groups

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 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!


 


Let’s 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.)

REAL NUMBER

The set of real number is a group under addition.

Addition satisfy all group property.

11)the set of real numbers is not a group under multiplication because the element 0 has no inverse in that group, as division by 0 does not make any sense. However, if you remove 0 from the set of real numbers then the resulting set will be a group with respect to multiplication.

NOTE-

Each element x of the set of non-zero real numbers R≠0 has an inverse element 1x under the operation of real number multiplication:


∀x∈R≠0:∃1x∈R≠0:x×1x=1=1x×x

The set of non zero real numbers (R*)is a group under multiplication.




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