Question

List the elements of 4Z/16Z and create the group table.

Answer 1

4Z={.......-8,-4,0,4,8,........}

16Z={....-32,16,0,16,32...}

Quotient group : $\frac{G}{H}=\left \{ x+H /x\epsilon G\right \}$ , H is identity

There fore : $\frac{4Z}{16Z}= \left \{ 0+16Z,4+16Z,8+16Z,12+16Z\right \}$

{Explanation for how above Quotient group is to be determined:

Since, by definition of Quotient Group as defined above, we have to take x from 4Z, where G=4Z, H=16Z.

So for, x=0, we get 0+16Z=16Z=H i. e. identity element og quotient group.

for, x=4, we get 4+16Z

for, x=8, we get 8+16Z

for, x=12,we get 12+16Z

for, x=16,we get 16+16Z=16Z (x+H=H iff x belongs to H)

for any x we take from 4Z, it will be one of above element

For example, if x=-4

Since we can write 16Z=16+16Z

So, adding - 4 in this gives, - 4+16+16Z=12+16Z.

Hence we get only four distinct elements.}

Group table:

(+)	16z	4+16Z	8+16Z	12+16Z
16Z	16Z	4+16Z	8+16Z	12+16Z
4+16Z	4+16Z	8+16Z	12+16Z	16Z
8+16Z	8+16Z	12+16Z	16Z	4+16Z
12+16Z	12+16Z	16Z	4+16Z	8+16Z

​​​​​

Explanation : to fill cell we added repective elemets.

To fill last right corner cell:

(12+16Z)+(12+16Z)=24+16Z {(x+H) +(y+H) =(x+y) +H}

=24+(-16+16Z) {Since x+H=H iff x is in H}

=8+16Z