Uniqueness of inverse element : Group Theory

Theorem : In a group G, inverse of  any element is unique. 

 

Proof :

             Let G be a group with the identity element e.

             

              Let a 𝞊 G.

          

             Since G is a group, a has an inverse element say b.

          

            Now we prove b is only the inverse element of a in G.

          

             In contrary, assume c is also the inverse of a in G.

             To prove a has unique inverse element , we have to prove b = c.

              

            Since b is inverse of a in G, we have a b = e = b a ............I

          

            Since c is inverse of a in G , we have a c = e = c a ..........II

      

            Now  c ( ab) = c(e) = c  ..........III    | since from I we have ab = e and ce=c |

            

            Also c ( ab ) = ( ca ) b = e b = b ...........IV  | since from II ca = e and eb=b |

            ∴ from III and IV , we have b = c

           

            i.e. the inverse of a in G is unique.

 

             Thus, the inverse  of any element of a group is unique.

                          

                    *** Hence The Proof ***

  Note : 

      

          * We denote the inverse of 'a' as  i)  a^-1  if  the binary operation is multiplication

                                                              ii)  -a  if the binary operation is addition.                                      

 * What is groupoid

 * What is monoid

 * What is abelian group

 * Uniqueness of identity element 

 *  Is aob = a^b is a binary operation on N?

 * Is aob = a is associative in S?

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