Normal Subgroups

1.                     HIGHER ALGEBRA

 ■  Normal Subgroups  ::-

Def :-    Let be a subgroup of a group G then H is said to be a Normal subgroup of G of aH=Ha  such that  a€G.

  ● Example :-

 1. Let H={1,-1}  & G={1,-1,i,-i} then H is a normal subgroup of G.

 Because H={1,-1} =-1+1,

           Q-1H={-1,1} =1+(-1),

                iH={-i,-i} =1+i,

               -iH={-i,i} = 1+(-1);

2. G=(Z,+), H=(2Z,+)

Then H♢G

because, a+H={a+h, h €H}

              

                         ={h+a : h €H

■■ Quotient Group ::-

 Def :- Let S={aH :a belong to G} .Where H is a normal subgroup of G. Let us defined a binary operation ○ on S by  aH○bH =abH for all aH, bH belong to S.

Then (S,○) is a group .

 ●● Because :-

 1. aH○bH=abH belong to S , where aH,bH belong to S

 ▪ S is Close with respect to '○'.

2.  aH○(bH○cH)=aH○(bc)H

                            =a (bc)H

                            =(ab)H○cH

                            =(aH○bH)○cH

• S is association in S.

 3.  eH =H & aH○H =aH○eH

                                =aeH

                                =aH

                   H○aH=eH○aH

                              =eaH

                              =aH.

• H is the identity element of S with respect to '○',