Question

In the group Z12, let H = 〈6〉 and N = 〈8〉. (a) List the elements of HN/N. (b) List the elements of H/(H ∩ N). (c) Define an ismorphism between HN/N and H/(H ∩ N).

Answer 1

Z12={0,1,2,3,4,5,6,7,8,9,10,11}

H=  〈6〉={6,12=0}

N= 〈8〉={8,16=4,12=0}

(A)

So HN={0,4,6,8}

So elements of HN/N are {{0,4,8},{6}}

Hence elements are {0,6} as 0,4,8 belongs to N

(B)

(H ∩ N)={0}

So H/(H ∩ N)={0,6}={0,6}

(C)

an ismorphism between HN/N and H/(H ∩ N)

Let define a homomorphism

β:HN/N --> H/(H ∩ N)

β(0)=0, And β(6)=6

β(6+6)=β(12)=β(0)=0=β(6)+β(6)=6+6=12

β(0+6)=β(6)=6=0+6=β(0)+β(6)

β(0+0)=β(0)=0=0+0=β(0)+β(0)

And cardinality of both the sets are same so it is a isomorphism.