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Elementary Properties of Groups

  • Last Updated : 25 Feb, 2021

Let the set G on which a binary operation o is defined from a group (G , o). G is a group if it satisfies the following 3 properties:

  • Associativity
  • Identity
  • Inverse

Properties of Groups :

Property-1
If a , b, c ∈ G  then, is a o b = a o c ⇒ b = c                 

Proof: –

Given a o b = a o c, for every a, b, c ∈  G  
Operating on the left with a-1, where a-1 ∈ G we have 
      a-1 o (a o b) = a-1 o (a o c) 
or  (a-1 o a) o b = (a-1 o a) o c         [using associative property]
or   e o b = e o c,                       [using inverse property]
or      b = c,                            [using identity property]

Note that a o b is also written as ab.



 This is known as the left cancellation law. 

Property-2: 
For every a  ∈ G , e o a = a = a o e, where e is the identity element. i.e. The left identity element is also the right identity element. 

Proof: – 

If a-1 be the left inverse of a, then 
              a-1 o (a o e) = (a-1 o a) o e           [using associative property]
 or         a-1 o (a o e) = e o e                     [using inverse property]
                                 = e                  [using identity property]
 or         a-1 o (a o e) = a-1 o a                   [using inverse property]
 i.e.        a-1 o (a o e) = a-1 o a  

Hence, a o e = a by property-1 i.e. left cancellation law. thus we find that e is also the right identity element and so it is called only the identity element. 

Property-3: 
For every a  ∈ G , a-1 o a = e = a o a-1 i.e. the left inverse of an element is also its right inverse.

Proof: –  

        a-1 o (a o a-1) = (a-1 o a) o a-1    [using identity property]
      = e o a-1                                  [using inverse property]
      = a-1 o e                              [by property 2]
 i.e. a-1 o (a o a-1)= a-1 o e
Hence, a o a-1 = e, by left cancellation law. 

Thus, we find that the left inverse a-1 of element a is also its right inverse and so a-1 is called only the inverse of a.

Property-4: 
If a , b, c ∈ G  then, is b o a = c o a ⇒ b = c                



Proof: – 

Given a o b = a o c, for every a, b, c ∈  G  
Operating on the left with a-1, where a-1 ∈ G we have
       (b o a) o a-1 =  (c o a) o a-1
or      b o (a-1 o a)  = c o (a-1 o a)           [using associative property]
or      b o e = c o e,                           [using inverse property]
or      b = c,                                   [using identity property]

This is known as right cancellation law. 

Property-5: 
For every a , b ∈ G we have  (a o b)-1 = b-1 o a-1 i.e. The inverse of the product (or the composite) of two elements a, b of group G is the product (or composite) of the inverses of the two elements taken in the reverse order. 

Proof: –  

Let a-1 and b-1 be the inverses of a and b. 
Now,(a o b) o (b-1 o a-1) = a o (b o b-1)  o a-1        [using  associative property]
= a o e o a-1                                  [using inverse property]
= a o a-1                                       [using identity property]
= e                                                [using inverse property]
(a o b) o (b-1 o a-1) = e
Similarly, (b-1 o a-1) o ( a o b)= e

Therefore, by the definition of inverse b-1 o a-1 is the inverse of a o b. i.e.  (a o b)-1=b-1 o a-1 

This is known as the reversal rule.

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